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Research ArticleAnisotropic Hardy Spaces of Musielak-Orlicz Type withApplications to Boundedness of Sublinear Operators

Baode Li1 Dachun Yang2 and Wen Yuan2

1 School of Mathematics and System Sciences Xinjiang University Urumqi 830046 China2 School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex SystemsMinistry of Education Beijing 100875 China

Correspondence should be addressed to Dachun Yang dcyangbnueducn

Received 17 November 2013 Accepted 15 January 2014 Published 16 March 2014

Academic Editors A Agouzal A Fiorenza and A Ibeas

Copyright copy 2014 Baode Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Let 120593 R119899 times [0infin) rarr [0infin) be a Musielak-Orlicz function and 119860 an expansive dilation In this paper the authors introducethe anisotropic Hardy space of Musielak-Orlicz type 119867120593

119860(R119899) via the grand maximal function The authors then obtain some

real-variable characterizations of 119867120593119860(R119899) in terms of the radial the nontangential and the tangential maximal functions which

generalize the known results on the anisotropic Hardy space 119867119901119860(R119899) with 119901 isin (0 1] and are new even for its weighted variant

Finally the authors characterize these spaces by anisotropic atomic decompositions The authors also obtain the finite atomicdecomposition characterization of 119867120593

119860(R119899) and as an application the authors prove that for a given admissible triplet (120593 119902 119904)

if 119879 is a sublinear operator and maps all (120593 119902 119904)-atoms with 119902 lt infin (or all continuous (120593 119902 119904)-atoms with 119902 = infin) into uniformlybounded elements of some quasi-Banach spacesB then 119879 uniquely extends to a bounded sublinear operator from119867

120593

119860(R119899) toB

These results are new even for anisotropic Orlicz-Hardy spaces on R119899

1 Introduction

The theory of Hardy spaces on the Euclidean space R119899

plays an important role in various fields of analysis andpartial differential equations (see eg [1ndash5]) One of themost important applications of Hardy spaces is that theyare good substitutes of Lebesgue spaces when 119901 isin (0 1]For example when 119901 isin (0 1] it is well known that Riesztransforms are not bounded on 119871

119901

(R119899) however they arebounded on Hardy spaces 119867119901(R119899) Moreover there wereseveral efforts to extend classicalHardy spaces some ofwhichare weighted anisotropic Hardy spaces [6] associated withgeneral expansive dilations and 119860

infinMuckenhoupt weights

TheseHardy spaces include classical isotropicHardy spaces ofFefferman and Stein [1] parabolic Hardy spaces of Calderonand Torchinsky [7] and weighted Hardy spaces of Garcıa-Cuerva [8] as well as Stromberg and Torchinsky [5] asspecial cases Apart from their theoretical consideration suchanisotropic function spaces also play an important role inallowing evenmore general discrete dilation structures which

have originated from the theory of wavelets see for example[9 10]

On the other hand as a generalization of 119871119901(R119899) theOrlicz space was introduced by Birnbaum and Orlicz in[11] and Orlicz in [12] Since then the theory of the Orliczspaces themselves has been well developed and these spaceshave been widely used in many branches of analysis (seeeg [13ndash15]) Moreover as a development of the theory ofOrlicz spaces Orlicz-Hardy spaces and their dual spaces werestudied by Stromberg [16] and Janson [17] on R119899 and quiterecently Orlicz-Hardy spaces associated with divergenceform elliptic operators by Jiang and Yang [18]

Let A119902(R119899) with 119902 isin [1infin] denote the class of Muck-

enhoupt weights (see eg [19] for their definitions andproperties) and let 120593 be a growth function (see [20]) whichmeans that 120593 R119899 times [0infin) rarr [0infin) is a Musielak-Orlicz function such that 120593(119909 sdot) is an Orlicz function and120593(sdot 119905) is a Muckenhoupt A

infin(R119899) weight It is known that

Musielak-Orlicz functions are the natural generalization ofOrlicz functions that may vary in the spatial variables (see

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 306214 19 pageshttpdxdoiorg1011552014306214

2 The Scientific World Journal

eg [20ndash23]) Recently Ky [20] introduced a new Musielak-Orlicz Hardy space119867120593(R119899) via the grand maximal functionand established its atomic characterization It is knownthat 119867120593(R119899) generalizes both the Orlicz-Hardy space ofStromberg [16] and Janson [17] and the weightedHardy space119867119901

119908(R119899)with119908 isin A

infin(R119899) studied by Garcıa-Cuerva [8] and

Stromberg and Torchinsky [5] Recall that the motivation tostudy function spaces of Musielak-Orlicz type comes fromtheir applications to many branches of mathematics andphysics (see eg [20 23ndash27]) In [20] Ky further introducedthe BMO-type space BMO

120593(R119899) which was proven to be

the dual space of 119867120593(R119899) as an interesting application Kyproved that the class of pointwise multipliers for BMO(R119899)characterized by Nakai and Yabuta [28 29] is the dual spaceof 1198711(R119899) +119867log

(R119899) where119867log(R119899) denotes the Musielak-

Orlicz Hardy space related to the growth function

120593 (119909 119905) =119905

log (119890 + |119909|) + log (119890 + 119905) (1)

for all 119909 isin R119899 and 119905 isin [0infin) It is worth noticing that somespecial Musielak-Orlicz Hardy spaces appear naturally in thestudy of the products of functions in BMO(R119899) and 1198671(R119899)(see [25 26 30]) the endpoint estimates for the div-curllemma and the commutators of singular integral operators(see [25 30ndash32])

Moreover observe that a distribution inHardy spaces canbe represented as a (finite or infinite) linear combination ofatoms (see [33 34]) Then the boundedness of linear oper-ators in Hardy spaces can be deduced from their behavioron atoms in principle However Meyer et al [35 page 513]gave an example of 119891 isin 119867

1

(R119899) whose norm can not beachieved by its finite atomic decompositions via (1infin 0)-atoms Applying this Bownik [36] showed that there existsa linear functional defined on a dense subspace of 1198671(R119899)which maps all (1infin 0)-atoms into bounded scalars butyet can not extend to a bounded linear functional on thewhole 1198671(R119899) Let 119901 isin (0 1] and let 119904 be a nonnegativeinteger not less than 119899(1119901 minus 1) This implies that theuniform boundedness in some quasi-Banach space B of alinear operator 119879 on all (119901infin 119904)-atoms does not generallyguarantee the boundedness of 119879 from 119867

119901

(R119899) to B Thisphenomenon has also essentially already been observed byMeyer et al in [37 page 19] Motivated by [36] via usingthe Lusin function characterization of Hardy spaces119867119901(R119899)Yang and Zhou [38] proved that a B

120574-sublinear operator

119879 uniquely extends to a bounded B120574-sublinear operator

from 119867119901

(R119899) with 119901 isin (0 1] to some quasi-Banach spaceB if and only if 119879 maps all (119901 2 119904)-atoms into uniformlybounded elements of B Independently Meda et al [39]established anothermore general bounded criterion via usingthe grand maximal function characterization of 119867119901(R119899)precisely they proved that if 119879 is a linear operator and mapsall (119901 119902 119904)-atoms with 119902 lt infin or all continuous (119901infin 119904)-atoms into uniformly bounded elements of a Banach spaceB then 119879 uniquely extends to a bounded linear operatorfrom 119867

119901

(R119899) to B This result was further generalizedto the weighted anisotropic Hardy spaces in [6] weighted

anisotropic product Hardy spaces in [40] and especiallyHardy spaces of Musielak-Orlicz type by Ky in [20]

There are three goals in this paper First we introduceanisotropic Hardy spaces of Musielak-Orlicz type 119867120593

119860(R119899)

via grand maximal functions and characterize these spacesvia anisotropic atomic decompositions These Hardy spacesinclude classical Hardy spaces119867119901(R119899) of Fefferman and Stein[1] weighted anisotropic Hardy spaces of Bownik [6] andHardy spaces of Musielak-Orlicz type of Ky [20]

The second goal is to obtain some new real-variable char-acterizations of119867120593

119860(R119899) in terms of the radial the nontangen-

tial and the tangential maximal functions via some boundedestimates of the truncated maximal function pointwise orin anisotropic Musielak-Orlicz spaces which are motivatedby [9 Section 7] These real-variable characterizations of119867120593

119860(R119899) coincide with the known best results when119867120593

119860(R119899)

is the anisotropic Hardy space 119867119901119860(R119899) with 119901 isin (0 1] (see

[9 Theorem 71]) or new even in its weighted variantThe third goal is to generalize the result ofMeda et al [39]

to the present setting More precisely we prove the existenceof finite atomic decompositions achieving the norm in densesubspaces of 119867120593

119860(R119899) As an application we prove that for a

given admissible triplet (120593 119902 119904) (seeDefinition 30 below) if119879is a B

120574-sublinear operator and maps all (120593 119902 119904)-atoms with

119902 lt infin (or all continuous (120593 119902 119904)-atoms with 119902 = infin) intouniformly bounded elements of some quasi-Banach spacesB then 119879 uniquely extends to a bounded B

120574-sublinear

operator from 119867120593

119860(R119899) to B These results are new even for

the anisotropic Hardy-Orlicz spaces on R119899This paper is organized as follows In Section 2 we first

recall some notation and definitions concerning Musielak-Orlicz functions expansive dilations and Muckenhouptweights Then we introduce the anisotropic Hardy spaces ofMusielak-Orlicz type119867120593

119860(R119899) via grand maximal functions

and some basic properties of these spaces are also presentedIn Section 3 we obtain some new real-variable characteri-zations of 119867120593

119860(R119899) via the radial the nontangential and the

tangential maximal functions Section 4 is devoted to gener-alizing the Calderon-Zygmund decomposition associated toweighted anisotropic Hardy spaces in [6] to the more generalspaces119867120593

119860(R119899) Applying this in Section 5 we introduce the

anisotropic atomic Hardy spaces of Musielak-Orlicz type119867120593119902119904

119860(R119899) for any admissible triplet (120593 119902 119904) and further

prove that for any admissible triplet (120593 119902 119904)

119867120593

119860(R119899

) = 119867120593119902119904

119860(R119899

) (2)

with equivalent norms (see Theorem 40 below) Moreoverin Section 61 we prove that sdot

119867120593119902119904

119860fin(R119899)and sdot

119867120593

119860(R119899) are

equivalent quasinorms on 119867120593119902119904

119860fin(R119899

) when 119902 lt infin and on119867120593119902119904

119860fin(R119899

) cap C(R119899) when 119902 = infin where 119867120593119902119904119860fin(R

119899

) denotesthe space of all finite linear combinations of multiples of(120593 119902 119904)-atoms In Section 62 we obtain criteria for bound-edness of sublinear operators in 119867

120593

119860(R119899) (see Theorem 44

below) The results in Section 6 are also new even for theanisotropic Hardy-Orlicz spaces on R119899

Finally we make some conventions on notation Let N =

1 2 and let Z+= 0 cup N Denote by S(R119899) the space

The Scientific World Journal 3

of all Schwartz functions andS1015840(R119899) the space of all tempereddistributions For any 120572 = (120572

1 120572

119899) isin Z119899

+ |120572| = 120572

1+

sdot sdot sdot + 120572119899and 120597

120572

= (1205971205971199091)1205721 sdot sdot sdot (120597120597119909

119899)120572119899 Throughout the

whole paper we denote by 119862 a positive constant which isindependent of the main parameters but it may vary fromline to line The symbol 119863 ≲ 119865means that 119863 le 119862119865 If 119863 ≲ 119865

and 119865 ≲ 119863 we then write 119863 sim 119865 If 119864 is a subset of R119899 wedenote by 120594

119864its characteristic function For any 119886 isin R lfloor119886rfloor

denotes themaximal integer not larger than 119886

2 Anisotropic Hardy Spaces ofMusielak-Orlicz Type

In this section we introduce anisotropic Hardy spaces ofMusielak-Orlicz type via grand maximal functions and giveout some basic properties

First let us recall some notation for Orlicz functions seefor example [20] A function 120601 [0infin) rarr [0infin) is calledan Orlicz function if it is nondecreasing and 120601(0) = 0 120601(119905) gt0 if 119905 gt 0 and lim

119905rarrinfin120601(119905) = infin Observe that differently

from the classical Orlicz functions being convex the Orliczfunctions in this papermay not be convex AnOrlicz function120601 is said to be of lower (resp upper) type119901with119901 isin (minusinfininfin)if there exists a positive constant119862 such that for all 119905 isin [0infin)

and 119904 isin (0 1) (resp 119904 isin [1infin))

120601 (119904119905) le 119862119904119901

120601 (119905) (3)

Given the function 120593 R119899 times [0infin) rarr [0infin) such thatfor any 119909 isin R119899 120593(119909 sdot) is an Orlicz function 120593 is said to beof uniformly lower (resp upper) type 119901 with 119901 isin (minusinfininfin)if there exists a positive constant 119862 such that for all 119909 isin R119899119905 isin (0infin) and 119904 isin (0 1) (resp 119904 isin [1infin))

120593 (119909 119904119905) le 119862119904119901

120593 (119909 119905) (4)

120593 is said to be of positive uniformly lower (resp upper) typeif it is of uniformly lower (resp upper) type 119901 for some 119901 isin

(0infin) Let

119894 (120593) = sup 119901 isin (minusinfininfin)

120593 is of uniformly lower type 119901

119868 (120593) = inf 119901 isin (minusinfininfin)

120593 is of uniformly upper type 119901

(5)

denote the uniformly critical lower type and the critical uppertype of the function 120593 respectively

Now we recall the notion of expansive dilations on R119899see [9] A real 119899 times 119899 matrix 119860 is called an expansive dilationshortly a dilation if min

120582isin120590(119860)|120582| gt 1 where 120590(119860) denotes

the set of all eigenvalues of 119860 Let 120582minusand 120582

+be two positive

numbers such that

1 lt 120582minuslt min |120582| 120582 isin 120590 (119860) le max |120582| 120582 isin 120590 (119860) lt 120582

+

(6)

In the case when119860 is diagonalizable overC we can even take120582minus= min|120582| 120582 isin 120590(119860) and 120582

+= max|120582| 120582 isin 120590(119860)

Otherwise we need to choose them sufficiently close to theseequalities according to what we need in our arguments

It was proved in [9 Lemma 22] that for a given dilation119860 there exist an open ellipsoid Δ and 119903 isin (1infin) such thatΔ sub 119903Δ sub 119860Δ and one can additionally assume that |Δ| = 1where |Δ|denotes the 119899-dimensional Lebesguemeasure of theset Δ Let 119861

119896= 119860

119896

Δ for 119896 isin Z Then 119861119896is open 119861

119896sub 119903119861

119896sub

119861119896+1

and |119861119896| = 119887

119896 Throughout the whole paper let 120590 be theminimal integer such that 119903120590 ge 2 and for any subset 119864 of R119899let 119864∁ = R119899 119864 Then for all 119896 119895 isin Zwith 119896 le 119895 it holds truethat

119861119896+ 119861

119895sub 119861

119895+120590 (7)

119861119896+ (119861

119896+120590)∁

sub (119861119896)∁

(8)

where 119864+119865 denotes the algebraic sums 119909+119910 119909 isin 119864 119910 isin 119865of sets 119864 119865 sub R119899

Definition 1 A quasinorm associated with an expansivematrix 119860 is a Borel measurable mapping 120588

119860 R119899 rarr [0infin)

for simplicity denoted by 120588 such that

(i) 120588(119909) gt 0 for all 119909 isin R 0(ii) 120588(119860119909) = 119887120588(119909) for all 119909 isin R119899 where 119887 = | det119860|(iii) 120588(119909 + 119910) le 119867[120588(119909) + 120588(119910)] for all 119909 119910 isin R119899 where

119867 isin [1infin) is a constant

In the standard dyadic case119860 = 2119868119899times119899

120588(119909) = |119909|119899 for all119909 isin R119899 is an example of homogeneous quasinorms associatedwith 119860 here and hereafter 119868

119899times119899always denotes the 119899 times 119899 unit

matrix and | sdot | the Euclidean norm in R119899It was proved in [9 Lemma 24] that all homogeneous

quasinorms associated with a given dilation119860 are equivalentTherefore for a given expansive dilation 119860 in what followsfor convenience we always use the step homogeneous quasi-norm 120588 defined by setting for all 119909 isin R119899

120588 (119909) = sum

119896isinZ

119887119896

120594119861119896+1119861119896

(119909) if 119909 = 0 or else 120588 (0) = 0 (9)

By (7) and (8) we know that for all 119909 119910 isin R119899

120588 (119909 + 119910) le 119887120590

(max 120588 (119909) 120588 (119910)) le 119887120590 [120588 (119909) + 120588 (119910)] (10)

see [9 page 8] Moreover (R119899 120588 119889119909) is a space of homoge-neous type in the sense of Coifman andWeiss [41] where 119889119909denotes the 119899-dimensional Lebesgue measure

Definition 2 Let 119901 isin [1infin) A function 120593 R119899 times [0infin) rarr

[0infin) is said to satisfy the uniform anisotropic Muckenhouptcondition A

119901(119860) denoted by 120593 isin A

119901(119860) if there exists a

positive constant 119862 such that for all 119905 isin (0infin) when 119901 isin

(1infin)

sup119909isinR119899

sup119896isinZ

119887minus119896

int119909+119861119896

120593 (119910 119905) 119889119910

times 119887minus119896

int119909+119861119896

[120593(119910 119905)]minus1(119901minus1)

119889119910

119901minus1

le 119862

(11)


and when 119901 = 1


sup119896isinZ

119887minus119896

int119909+119861119896

120593 (119910 119905) 119889119910ess sup119910isin119909+119861119896

[120593(119910 119905)]minus1

le 119862

(12)

The minimal constant 119862 as above is denoted by 119862119901119860119899

(120593)Define A

infin(119860) = ⋃

1le119901ltinfinA119901(119860) and

119902 (120593) = inf 119902 isin [1infin) 120593 isin A119902(119860) (13)

If 120593 isin Ainfin(119860) is independent of 119905 isin [0infin) then 120593

is just an anisotropic Muckenhoupt 119860infin(119860) weight in [42]

Obviously 119902(120593) isin [1infin) If 119902(120593) isin (1infin) by a discussionsimilar to [6 page 3072] it is easy to know 120593 notin A

119902(120593)(119860)

Moreover there exists a 120593 isin (cap119902gt1

A119902(119860)) A

1(119860) such

that 119902(120593) = 1 see Johnson and Neugebauer [43 page 254Remark]

Now we introduce anisotropic growth functions

Definition 3 A function 120593 R119899 times [0infin) rarr [0infin) is calledan anisotropic growth function if

(i) the function 120593 is an anisotropic Musielak-Orliczfunction that is

(a) the function 120593(119909 sdot) [0infin) rarr [0infin) is anOrlicz function for all 119909 isin R119899

(b) the function 120593(sdot 119905) is a Lebesgue measurablefunction for all 119905 isin [0infin)

(ii) the function 120593 belongs to Ainfin(119860)

(iii) the function 120593 is of positive uniformly lower type 119901for some 119901 isin (0 1] and of uniformly upper type 1

Given a growth function 120593 let

119898(120593) = lfloor(119902 (120593)

119894 (120593)minus 1)

ln 119887ln 120582

minus

rfloor (14)

Clearly

120593 (119909 119905) = 119908 (119909)Φ (119905) forall119909 isin R119899

119905 isin [0infin) (15)

is an anisotropic growth function if 119908 is a classical or ananisotropic 119860

infinMuckenhoupt weight (cf [42]) and Φ of

positive lower type 119901 for some 119901 isin (0 1] and of uppertype 1 More examples of growth functions can be found in[20 22 30 32]

Remark 4 By Lemma 11 below (see also [20 Lemma 41])without loss of generality we may always assume that ananisotropic growth function 120593 is of positive uniformly lowertype 119901 for some 119901 isin (0 1] and of uniformly upper type 1 suchthat 120593(119909 sdot) is continuous and strictly increasing for all given119909 isin R119899

Throughout the whole paper we always assume that 120593is an anisotropic growth function Recall that the Musielak-Orlicz-type space 119871120593(R119899) is defined to be the set of allmeasurable functions 119891 such that for some 120582 isin (0infin)

intR119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 lt infin (16)

with the Luxembourg (or called the Luxembourg-Nakano)(quasi)norm

10038171003817100381710038171198911003817100381710038171003817119871120593(R119899) = inf 120582 isin (0infin) int

R119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 le 1

(17)

For119898 isin N let

S119898(R119899

) = 120601 isin S (R119899

)

sup119909isinR119899|120572|le119898+1

[1 + 120588(119909)]119898+2 1003816100381610038161003816120597

120572

120601 (119909)1003816100381610038161003816 le 1

(18)

In what follows for 120593 isin S(R119899) 119896 isin Z and 119909 isin R119899 let120593119896(119909) = 119887

119896

120593(119860119896

119909)For 119891 isin S1015840(R119899) the nontangential grand maximal

function 119891lowast119898of 119891 is defined by setting for all 119909 isin R119899

119891lowast

119898(119909) = sup

120601isinS119898(R119899)

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816119891 lowast 120601119896 (119910)1003816100381610038161003816 (19)

If119898 = 119898(120593) we then write 119891lowast instead of 119891lowast119898

Definition 5 For any119898 isin N and anisotropic growth function120593 the anisotropic Hardy space 119867120593

119898119860(R119899) of Musielak-Orlicz

type is defined to be the set of all 119891 isin S1015840(R119899) such that119891lowast

119898isin 119871

120593

(R119899) with the (quasi)norm 119891119867120593

119898119860(R119899) = 119891

lowast

119898119871120593(R119899)

When119898 = 119898(120593)119867120593

119898119860(R119899) is denoted simply by119867120593

119860(R119899)

Observe that when 119860 = 2119868119899times119899

and 120593 is as in (15) with aMuckenhoupt weight 119908 and an Orlicz function Φ the aboveHardy spaces119867120593

119860(R119899) are just weighted Hardy-Orlicz spaces

which include classical Hardy-Orlicz spaces of Janson [44](119908 equiv 1 in this context) and classical weighted Hardy spacesof Garcıa-Cuerva [8] as well as Stromberg and Torchinsky[5] (Φ(119905) = 119905

119901 for all 119905 isin [0infin) in this context) see also[19 45 46] When 120593 is as in (15) with Φ(119905) = 119905

119901 for all119905 isin [0infin) the above Hardy spaces119867120593

119860(R119899) become weighted

anisotropic Hardy spaces (see [6]) and more generally whenΦ is an Orlicz function these Hardy spaces are new

Now let us give some basic properties of119867120593119898119860

(R119899)

Proposition 6 For 119898 isin N it holds true that 119867120593119898119860

(R119899) sub

S1015840(R119899) and the inclusion is continuous

Proof Let 119891 isin 119867120593

119898119860(R119899) For any 120601 isin S(R119899) and 119909 isin 119861

0

we have ⟨119891 120601⟩ = 119891 lowast 120595119909(119909) where 120595

119909(119910) = 120601(119909 minus 119910) for all

119910 isin R119899


By Definition 1 we see that

sup119909isin1198610 119910isinR

119899

1 + 120588 (119910)

1 + 120588 (119909 minus 119910)le 119887

2120590

(20)

Therefore it holds true that

1003816100381610038161003816⟨119891 120601⟩1003816100381610038161003816 =

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

1003816100381610038161003816100381610038161003816100381610038161003816

119891 lowast (120595119909

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

) (119909)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)inf119909isin1198610

119891lowast

119898(119909)

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)

100381710038171003817100381710038171205941198610

10038171003817100381710038171003817

minus1

119871120593(R119899)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(21)

This implies that119891 isin S1015840(R119899) and the inclusion is continuouswhich completes the proof of Proposition 6

Using Proposition 6 with an argument similar to that of[20 Proposition 52] we have the following conclusion thedetails being omitted

Proposition 7 Let 119898 isin N and let 120593 be an anisotropic growthfunction Then119867120593

119898119860(R119899) is complete

3 Characterizations of 119867120593119860(R119899) via

Maximal Functions

The goal of this section is to establish somemaximal functioncharacterizations of119867120593

119860(R119899) Let us begin with the notions of

anisotropic variants of the radial the nontangential and thetangential maximal functions

Definition 8 Let 120595 isin S(R119899) with intR119899120595(119909)119889119909 = 0 The

anisotropic radial the nontangential and the tangential max-imal functions of 119891 associated to 120595 are defined respectivelyby setting for all 119909 isin R119899

M0

120595119891 (119909) = sup

119896isinZ

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

M120595119891 (119909) = sup

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119879119873

120595119891 (119909) = sup

119896isinZ


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

[1 + 120588 (119860minus119896 (119909 minus 119910))]119873 119873 isin Z

+

(22)

Theorem 9 Let 120593 be an anisotropic growth function and 120595 isin

S(R119899) with intR119899120595(119909)119889119909 = 0 Then for any 119891 isin S1015840(R119899) the

following are equivalent

119891 isin 119867120593

119860(R119899

) (23)

119879119873

120595119891 isin 119871

120593

(R119899

) 119873 gt[119902 (120593)]

2

119894 (120593) (24)

M120595119891 isin 119871

120593

(R119899

) (25)

M0

120595119891 isin 119871

120593

(R119899

) (26)

Moreover for sufficiently large119898 there exist positive constants1198621 1198622 1198623 and 119862

4 independent of 119891 isin 119867

120593

119860(R119899) such that

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

=1003817100381710038171003817119891lowast

119898

1003817100381710038171003817119871120593(R119899) le 119862110038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198622

10038171003817100381710038171003817M0

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198623

10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198624

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(27)

The approach we use to proveTheorem 9 is motivated byBownik [9 Theorem 71] First we need the following twolemmas which come from [5 pages 7-8] and [20 Lemma41(ii)]

In what follows for any set 119864 and 119905 isin [0infin) let

120593 (119864 119905) = int119864

120593 (119909 119905) 119889119909 (28)

Lemma 10 Let 119902 isin [1infin) and 120593 isin A119902(119860) Then there exists

a positive constant 119862 such that for all 119909 isin R119899 119896 isin Z 119864 sub

(119909 + 119861119896) and 119905 isin (0infin)

120593 (119909 + 119861119896 119905)

120593 (119864 119905)le 119862

1003816100381610038161003816119909 + 1198611198961003816100381610038161003816119902

|119864|119902

(29)

Lemma 11 Let 120593 be an anisotropic growth function For all(119909 119905) isin R119899 times [0infin) 120593(119909 119905) = int

119905

0

(120593(119909 119904)119904)119889119904 is also ananisotropic growth function which is equivalent to 120593 moreover120593(119909 sdot) for any given 119909 isin R119899 is continuous and strictlyincreasing

We now recall some Peetre-type maximal functions from[9] These maximal functions are obtained via the truncationwith an additional extra decay term Namely for an integer119870 representing the truncation level and a real nonnegativenumber 119871 representing the decay level any 119909 isin R119899 and 119896 isin Zwe define

119898119870119871

(119909 119896) = [max 1 120588 (119860minus119870119909)]119871

(1 + 119887minus119896minus119870

)119871 (30)

and the following Peetre-type radial the nontangential thetangential the radial grand and the nontangential grandmaximal functions

M0(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

119898119870119871

(119909 119896)

M(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)

119879119873(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

[1 + 120588 (119860minus119896 (119909 minus 119910))]119873

119898119870119871

(119910 119896)

119873 isin Z+

1198910lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M0(119870119871)

120595119891 (119909)

119891lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M(119870119871)

120595119891 (119909)

(31)

where S119898(R119899) is as in (18)


We need some technical lemmas To begin with let 119865

R119899timesZ rarr [0infin) be an arbitrary Borel measurable functionFor fixed 119895 isin Z and119870 isin Z cup infin themaximal function of 119865with aperture 119895 is defined by setting for all 119909 isin R119899

119865lowast119870

119895(119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119895+119896

119865 (119910 119896) (32)

It was shown in [9 page 42] that 119865lowast119870119895

is lower semicontin-uous namely 119909 isin R119899 119865

lowast119870

119895(119909) gt 120582 is open for any

120582 isin (0infin)We have the following Lemma 12 associated to119865lowast119870

119895which

is a uniformly weighted analogue of [9 Lemma 72]

Lemma 12 Let 119902 isin [1infin) and 120593 isin A119902(119860) Then there exists a

positive constant119862 such that for any 120582 119905 isin [0infin) and 119895 isin Z+

120593 (119909 isin R119899 119865lowast119870119895

(119909) gt 119905 120582)

le 1198621198871199022119895

120593 (119909 isin R119899 119865lowast1198700

(119909) gt 119905 120582)

(33)

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 le 119862119887

1199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909 (34)

Proof For any 119905 isin [0infin) let Ω = 119909 isin R119899 119865lowast1198700

(119909) gt 119905For any 119909 isin R119899 satisfying 119865lowast119870

119895(119909) gt 119905 there exist 119896 le 119870

and 119910 isin 119909 + 119861119896+119895

such that 119865(119910 119896) gt 119905 Clearly 119910 + 119861119896sub Ω

Moreover by (7) and 119895 isin Z+ we find that

119910 + 119861119896sub 119909 + 119861

119896+119895+ 119861

119896sub 119909 + 119861

119896+119895+120590 (35)

From this and 120593 isin A119902(119860) with Lemma 10 it follows that

119887minus119902(119895+120590)

120593 (119909 + 119861119896+119895+120590

120582) le 1198621120593 (119910 + 119861

119896 120582) (36)

Consequently by this and 119910+119861119896sub Ωcap (119909 +119861

119896+119895+120590) we have

120593 (Ω cap (119909 + 119861119896+119895+120590

) 120582) ge 120593 (119910 + 119861119896 120582)

ge 119862minus1

1119887minus119902(119895+120590)

times 120593 (119909 + 119861119896+119895+120590

120582)

(37)

which implies that

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(38)

where M120593(sdot120582)

denotes the centered Hardy-Littlewood maxi-mal function associated to themeasure 120593(119909 120582)119889119909 namely forall 119909 isin R119899

M120593(sdot120582)

119891 (119909) = sup119898isinZ

1

120593 (119909 + 119861119898 120582)

times int119909+119861119898

1003816100381610038161003816119891 (119910)1003816100381610038161003816 120593 (119910 120582) 119889119910

(39)

Thus

119909 isin R119899

119865lowast119870

119895(119909) gt 119905

sub 119909 isin R119899

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(40)

From this and the weak-119871119902(R119899 120593(119909 120582)119889119909) boundedness ofM120593(sdot120582)

with 120593 isin A119902(119860) it is easy to deduce (33)

Next we prove (34) By Lemma 11 we know that

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 sim int

R119899int

119865lowast119870

119895(119909)

0

120593 (119909 119905)119889119905

119905119889119909

sim int

infin

0

int119909isinR119899119865lowast119870

119895(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

(41)

which together with (33) further implies that

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲ 1198871199022119895

int

infin

0

int119909isinR119899119865lowast119870

0(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

sim 1198871199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

(42)

which is desired This finishes the proof of Lemma 12

The following Lemma 13 is just [20 Lemma 41(i)]

Lemma 13 Let 120593 be an anisotropic growth function Thenthere exists a positive constant 119862 such that for all (119909 119905

119895) isin

R119899 times [0infin) with 119895 isin N

120593(119909

infin

sum

119895=1

119905119895) le 119862

infin

sum

119895=1

120593 (119909 119905119895) (43)

The following Lemma 14 extends [9 Lemma 75] to thesetting of anisotropic Musielak-Orlicz function spaces

Lemma 14 Let 120595 isin S(R119899) let 120593 be an anisotropic growthfunction and let 119873 isin ([119902(120593)]

2

119894(120593)infin) Then there exists apositive constant 119862 such that for all 119870 isin Z 119871 isin [0infin) and119891 isin S1015840(R119899)

10038171003817100381710038171003817119879119873(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

le 11986210038171003817100381710038171003817M(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

(44)

Proof For any 119891 isin S1015840(R119899) 120595 isin S(R119899) 119870 isin Z and 119871 isin

[0infin) consider a function 119865 R119899 times Z rarr [0infin) given bysetting for all (119910 119896) isin R119899 times Z

119865 (119910 119896) =

1003816100381610038161003816119891 lowast 120595119896 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)(45)

with 119898119870119871

being as in (30) Fix 119909 isin R119899 and 119873 isin

([119902(120593)]2

119894(120593)infin) If 119896 le 119870 and 119909 minus 119910 isin 119861119896 then

119865 (119910 119896) [max 1 120588 (119860minus119896 (119909 minus 119910))]minus119873

le 119865lowast119870

0(119909) (46)

where 119865lowast1198700

is as in (32) If 119896 le 119870 and 119909minus119910 isin 119861119896+119895+1

119861119896+119895

forsome 119895 isin Z

+ then


le 119887minus119895119873

119865lowast119870

119895(119909)

(47)


where 119865lowast119870119895

is as in (32) By taking supremum over all 119910 isin R119899

and 119896 le 119870 we obtain

119879119873(119870119871)

120595119891 (119909) le

infin

sum

119895=0

119887minus119895119873

119865lowast119870

119895(119909) (48)

Moreover since 119873 isin ([119902(120593)]2

119894(120593)infin) we choose 119901 lt 119894(120593)

large enough and 119902 gt 119902(120593) small enough such that119873119901minus 1199022 gt0 Therefore from this (48) Lemma 13 the uniformly lowertype 119901 of 120593 and Lemma 12 it follows that

intR119899120593 (119909 119879

119873(119870119871)

120595119891 (119909)) 119889119909

le

infin

sum

119895=0

119887minus119895119873119901

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲

infin

sum

119895=0

119887minus119895(119873119901minus119902

2)

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

≲ intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(49)

which implies (44) This finishes the proof of Lemma 14

The following Lemmas 16 and 18 are just [9 Lemmas 75and 76] respectively

Lemma 15 Suppose 120595 isin S(R119899) with intR119899120595(119909) 119889119909 = 0 Then

for any given 119873 119871 isin [0infin) there exist a positive integer 119898and a positive constant119862 such that for all119891 isin S1015840(R119899) integers119870 isin Z

+and 119909 isin R119899

119891lowast0(119870119871)

119898(119909) le 119862119879

119873(119870119871)

120595119891 (119909) (50)

Lemma 16 Let 120595 isin S(R119899) with intR119899120595(119909)119889119909 = 0 and 119891 isin

S1015840(R119899) Then for every 119872 isin (0infin) there exists 119871 isin (0infin)

such that for all 119909 isin R119899

M(119870119871)

120595119891 (119909) le 119862[max 1 120588 (119909)]minus119872 (51)

where 119862 is a positive constant depending on 119870119872 119871 isin Z+ 119860

and 120595 but independent of 119891 and 119909

The following Lemma 17 is just [9 Proposition 310] and[6 Proposition 211]

Lemma 17 There exists a positive constant 119862 such that foralmost every 119909 isin R119899119898 isin N and 119891 isin 119871

1

loc(R119899

) capS1015840(R119899)

119891 (119909) le 119891lowast

119898(119909) le 119862119891

lowast0

119898(119909) le 119862M

119860119891 (119909) (52)

where 119891lowast0119898(119909) = sup

120601isinS119898(R119899)sup

119896isinZ|119891 lowast 120601119896(119909)| for all 119909 isin R119899

and M119860denotes the anisotropic Hardy-Littlewood maximal

operator defined by setting for all 119909 isin R119899

M119860119891 (119909) = sup

119909isin119861119861isinB

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (53)

The following lemma comes from [22 Corollary 28] witha slight modification the details being omitted

Lemma 18 Let 120593 be an anisotropic Musielak-Orlicz functionwith uniformly lower type 119901minus

120593and uniformly upper type 119901+

120593

satisfying 119902(120593) lt 119901minus

120593le 119901

+

120593lt infin where 119902(120593) is as in (13)

Then the Hardy-Littlewood maximal operatorM119860is bounded

on 119871120593(R119899)

Proof of Theorem 9 Obviously (23)rArr (25)rArr (26) Let 120593 bean anisotropic growth function and let 120595 isin S(R119899) satisfyintR119899120595(119909)119889119909 = 0 By (50) of Lemma 15 with 119871 = 0 and 119873 isin

([119902(120593)]2

119894(120593)infin) we know that there exists a positive integer119898 such that for all 119891 isin S1015840(R119899) 119909 isin R119899 and integers119870 isin Z

+

119891lowast0(1198700)

119898(119909) ≲ 119879

119873(1198700)

120595119891 (119909) (54)

From this and Lemma 14 it follows that for all 119891 isin S1015840(R119899)and119870 isin Z

+10038171003817100381710038171003817119891lowast0(1198700)

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

(55)

As119870 rarr infin by the monotone convergence theorem and thecontinuity of 120593(119909 sdot) (see Lemma 11) we have

10038171003817100381710038171003817119891lowast0

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

(56)

which together with Lemma 17 implies that (25)rArr (24)rArr(23) It remains to prove (26)rArr (23)

SupposeM0

120595119891 isin 119871

120593

(R119899) By Lemma 16 we find some 119871 isin(0infin) such that (51) holds true which implies thatM(119870119871)

120595119891 isin

119871120593

(R119899) for all 119870 isin Z+ By Lemmas 14 and 15 we find 119898 isin N

such that

intR119899120593 (119909 119891

lowast0(119870119871)

119898(119909)) 119889119909

le 1198621intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(57)

with a positive constant1198621being independent of119870 isin Z

+ For

any given 119870 isin Z+ let

Ω119870= 119909 isin R

119899

1198910lowast(119870119871)

119898(119909) le 119862

2M(119870119871)

120595119891 (119909) (58)

where 1198622= [2119862

1]1119901 with 119901 isin (0 119894(120593)) We claim that

intR119899120593 (119909M

(119870119871)

120595119891 (119909)) le 2int

Ω119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909 (59)

Indeed by (57) the uniformly lower type 119901 of 120593 and119862minus11990121198621=

12 we have

intΩ∁

119870

120593 (119909M(119870119871)

120595(119909)) lt 119862

minus119901

2intΩ∁

119870

120593 (119909 1198910lowast(119870119871)

119898(119909)) 119889119909

le 119862minus119901

21198621intR119899120593 (119909M

(119870119871)

120595(119909)) 119889119909

(60)


Moreover for any 119909 isin Ω119870and 119901 isin (0 119894(120593)) we choose 119902 isin

(0 119901) small enough such that 1119902 gt 119902(120593) where 119902(120593) is as in(13) and by [9 page 48 (716)] we know that there exists aconstant 119862

3isin (1infin) such that for all integers 119870 isin Z

+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)

Furthermore from the fact that 120593 is of uniformly upper type1 and positive lower type 119901 with 119901 lt 119894(120593) it follows that120593(119909 119905) = 120593(119909 119905

1119902

) is of uniformly upper 1119902 and lower type119901119902 Consequently using (59) (61) and Lemma 18 with 120593 weobtain

intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)

where 1198624depends on 119871 isin [0infin) but is independent of

119870 isin Z+ This inequality is crucial since it gives a bound of

the nontangential maximal function by the radial maximalfunction in 119871120593(R119899)

Since M(119870119871)

120595119891(119909) converges pointwise and monotoni-

cally to M120595119891(119909) for all 119909 isin R119899 as 119870 rarr infin it follows

that M120595119891 isin 119871

120593

(R119899) by (62) the continuity of 120593(119909 sdot)(see Lemma 11) and the monotone convergence theoremTherefore by choosing 119871 = 0 and using (62) the continuity of120593(119909 sdot) and themonotone convergence theorem we concludethat M

120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)

where now the positiveconstant 119862

4corresponds to 119871 = 0 and is independent

of 119891 isin S1015840(R119899) Combining this (56) and Lemma 17 weobtain the desired conclusion and hence complete the proofof Theorem 9

4 Calderoacuten-Zygmund Decompositions

In this section by using the Calderon-Zygmund decomposi-tion associated with grand maximal functions on anisotropicR119899 established in [6] we obtain some bounded estimates on119867120593

119860(R119899) We follow the constructions in [2 6]Throughout this section we consider a tempered distribu-

tion 119891 so that for all 120582 119905 isin (0infin)

int119909isinR119899119891lowast

119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)

where119898 ge 119898(120593) is some fixed integer For a given 120582 isin (0infin)let

Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)

By referring to [6 page 3081] we know that there exist apositive constant 119871 independent of Ω and 119891 a sequence119909119895119895

sub Ω and a sequence of integers ℓ119895119895

such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861

ℓ119895minus2120590) = 0 forall119894 119895 with 119894 = 119895 (66)

(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)

Here and hereafter for a set 119864 119864 denotes its cardinalityFix 120579 isin S(R119899) such that supp 120579 sub 119861

120590 0 le 120579 le 1 and 120579 equiv 1

on 1198610 For each 119895 and all 119909 isin R119899 define 120579

119895(119909) = 120579(119860

minusℓ119895(119909 minus

119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By

(65) and (69) for any 119909 isin Ω we have 1 le sum119895120579119895(119909) le 119871 For

every 119894 and all 119909 isin R119899 define

120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861

ℓ119894minus2120590by (66) and sum

119894120577119894= 120594

Ω Therefore the family 120577

119894119894

forms a smooth partition of unity onΩLet 119904 isin Z

+be some fixed integer and let P

119904(R119899) denote

the linear space of polynomials of degrees not more than 119904For each 119894 and 119875 isin P

119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot

119894) is a finite dimensional Hilbert space Let

119891 isin S1015840(R119899) For each 119894 since 119891 induces a linear functionalon P

119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577

119894⟩ by the Riesz

lemma we know that there exists a unique polynomial 119875119894isin

P119904(R119899) such that for all 119876 isin P

119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)

For every 119894 define a distribution 119887119894= (119891 minus 119875

119894)120577119894

We will show that for suitable choices of 119904 and 119898 theseries sum

119894119887119894converges in S1015840(R119899) and in this case we define

119892 = 119891 minus sum119894119887119894in S1015840(R119899)

Definition 19 The representation 119891 = 119892 + sum119894119887119894 where 119892 and

119887119894are as above is called a Calderon-Zygmund decomposition

of degree 119904 and height 120582 associated with 119891lowast119898


The remainder of this section consists of a series oflemmas In Lemmas 20 and 21 we give some properties ofthe smooth partition of unity 120577

119894119894 In Lemmas 22 through

25 we derive some estimates for the bad parts 119887119894119894 Lemmas

26 and 27 give some estimates over the good part 119892 FinallyCorollary 28 shows the density of 119871119902

120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593

119860(R119899) where 119902 isin (119902(120593)infin)Lemmas 20 through 23 are essentially Lemmas 43

through 46 of [9] the details being omitted

Lemma20 There exists a positive constant1198621 depending only

on119898 such that for all 119894 and ℓ le ℓ119894

sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)

Lemma 21 There exists a positive constant1198622 independent of

119891 and 120582 such that for all 119894


1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)

Lemma 22 There exists a positive constant 1198623 independent

of 119891 and 120582 such that for all 119894 and 119909 isin 119909119894+ 119861

ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)

Lemma 23 If 119898 ge 119904 ge 0 then there exists a positive constant1198624 independent of 119891 and 120582 such that for all 119905 isin Z

+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)

Lemma 24 If 119898 ge 119904 ge lfloor119902(120593) ln 119887(119894(120593) ln 120582minus)rfloor then there

exists a positive constant 1198625such that for all 119891 isin 119867

120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)

Moreover the series sum119894119887119894converges in119867120593

119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)

where 119871 is as in (69)

Proof By Lemma 22 we know that

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)

Notice that 119904 ge lfloor119902(120593) ln 119887(119894(120593) ln 120582minus)rfloor implies that

119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901

gt 1 for sufficient small 120578 gt 0 and sufficientlarge 119901 lt 119894(120593) Using Lemma 10 with 120593 isin A

119902(120593)+120578(119860)

Lemma 23 and the fact that 119891lowast119898(119909) gt 120582 for all 119909 isin 119909

119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)

which gives (75)By (75) and (69) we see that

intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)

which together with the completeness of 119867120593119898119860

(R119899) (seeProposition 7) implies that sum

119894119887119894converges in 119867120593

119898119860(R119899) So

by Proposition 6 we know that the series sum119894119887119894converges

in S1015840(R119899) and therefore (sum119894119887119894)lowast

119898le sum

119894(119887119894)lowast

119898 From this

and Lemma 13 we deduce (76) This finishes the proof ofLemma 24

Let 119902 isin [1infin] We denote by 119871119902

120593(sdot1)(R119899) the usually

anisotropic weighted Lebesgue space with the anisotropicMuckenhoupt weight 120593(sdot 1) Then we have the followingtechnical lemma (see [6 Lemma 48]) the details beingomitted

Lemma 25 If 119902 isin (119902(120593)infin] and 119891 isin 119871119902

120593(sdot1)(R119899) then

the series sum119894119887119894converges in 119871

119902

120593(sdot1)(R119899) and there exists a

positive constant 1198626 independent of 119891 and 120582 such that

sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)

The following conclusion is essentially [9 Lemma 49]the details being omitted

Lemma 26 If 119898 ge 119904 ge 0 and sum119894119887119894converges in S1015840(R119899) then

there exists a positive constant1198627 independent of119891 and120582 such

that for all 119909 isin R119899

119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)


Lemma 27 Let 119901 isin (119894(120593) 1] and 119902 isin (119902(120593)infin)

(i) If119898 ge 119904 ge lfloor119902(120593) ln 119887(119894(120593) ln 120582minus)rfloor and 119891 isin 119867

120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902

120593(sdot1)(R119899) and there exists a positive

constant 1198628 independent of 119891 and 120582 such that

intR119899[119892lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)

(ii) If 119898 isin N and 119891 isin 119871119902

120593(sdot1)(R119899) then 119892 isin 119871

infin

(R119899)and there exists a positive constant 119862

9 independent of

119891 and 120582 such that 119892119871infin(R119899) le 1198629120582

Proof Since 119891 isin 119867120593

119898119860(R119899) by Lemma 24 we know that

sum119894119887119894converges in 119867

120593

119898119860(R119899) and therefore in S1015840(R119899) by

Proposition 6 Then by Lemma 26 we have


119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)

where 119905119894(119909) is as in Lemma 26 Observe that 119898 ge

lfloor119902(120593) ln 119887(119894(120593) ln 120582minus)rfloor implies that (120582

minus)119898+1

gt 119887119902(120593) More-

over for any fixed 119909 isin 119909119894+ (119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

) with 119905 isin Z+

we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)

From this the 119871119902119902(120593)120593(sdot1)

(ℓ119902(120593)

)-boundedness of the vector-valuedmaximal functionM

119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)

Noticing that 119891lowast119898gt 120582 on Ω then for some 119901 isin (0 119894(120593))

we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)

On the other hand since 119891lowast119898le 120582 onΩ∁ for any 119909 isin Ω∁ using

120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)

Combining the above two estimates with (86) we obtain thedesired conclusion of Lemma 27(i)

Moreover notice that if 119891 isin 119871119902

120593(sdot1)(R119899) then 119892 and 119887

119894119894

are functions By Lemma 25sum119894119887119894converges in119871119902

120593(sdot1)(R119899) and

hence in S1015840(R119899) due to the fact that 119871119902120593(sdot1)

(R119899) sub S1015840(R119899) iscontinuous embedding (see [6 Lemma 28]) Write

119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)

By Lemma 21 and (69) we have |119892(119909)| ≲ 120582 for all 119909 isin Ω and|119892(119909)| = |119891(119909)| le 119891

lowast

119898(119909) le 120582 for almost every 119909 isin Ω∁ which

leads to 119892119871infin(R119899) ≲ 120582 and hence (ii) holds true This finishes

the proof of Lemma 27

Corollary 28 For any 119902 isin (119902(120593)infin) and 119898 ge

lfloor119902(120593) ln 119887(119894(120593) ln 120582minus)rfloor the subset 119867120593

119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593

119898119860(R119899) For any 120582 isin (0infin) let 119891 =

119892120582

+ sum119894119887120582

119894be the Calderon-Zygmund decomposition of 119891

of degree 119904 with lfloor119902(120593) ln 119887[119901 ln(120582minus)]rfloor le 119904 le 119898 and height

120582 associated with 119891lowast119898as in Definition 19 Here we rewrite 119892

and 119887119894in Definition 19 into 119892120582 and 119887120582

119894 respectively By (76) of

Lemma 24 we know that1003817100381710038171003817100381710038171003817100381710038171003817

sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)


and therefore119892120582 rarr 119891 in119867120593119898119860

(R119899) as120582 rarr infinMoreover byLemma 27(i) we see that (119892lowast

119898)120582

isin 119871119902

120593(sdot1)(R119899) which together

with Lemma 17 implies that119892120582 isin 119871119902120593(sdot1)

(R119899)This finishes theproof of Corollary 28

5 Atomic Characterizations of 119867120593119860(R119899)

In this section we establish the equivalence between119867120593119860(R119899)

and anisotropic atomic Hardy spaces of Musielak-Orlicz type119867120593119902119904

119860(R119899) (see Theorem 40 below)

LetB = 119861 = 119909 + 119861119896 119909 isin R119899 119896 isin Z be the collection

of all dilated balls

Definition 29 For any119861 isin B and 119902 isin [1infin] let 119871119902120593(119861) be the

set of all measurable functions 119891 supported in 119861 such that

10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=

sup119905isin(0infin)

[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin

119902 isin [1infin)

10038171003817100381710038171198911003817100381710038171003817119871infin(119861) lt infin 119902 = infin

(92)

It is easy to show that (119871119902120593(119861) sdot

119871119902

120593(119861)) is a Banach

space Next we introduce anisotropic atomic Hardy spaces ofMusielak-Orlicz type

Definition 30 We have the following definitions

(i) An anisotropic triplet (120593 119902 119904) is said to be admissibleif 119902 isin (119902(120593)infin] and 119904 isin Z

+such that 119904 ge 119898(120593) with

119898(120593) as in (14)

(ii) For an admissible anisotropic triplet (120593 119902 119904) a mea-surable function 119886 is called an anisotropic (120593 119902 119904)-atom if

(a) 119886 isin 119871119902120593(119861) for some 119861 isin B

(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904

(iii) For an admissible anisotropic triplet (120593 119902 119904) theanisotropic atomic Hardy space of Musielak-Orlicztype 119867120593119902119904

119860(R119899) is defined to be the set of all distri-

butions 119891 isin S1015840(R119899) which can be represented as asum ofmultiples of anisotropic (120593 119902 119904)-atoms that is119891 = sum

119895119886119895inS1015840(R119899) where 119886

119895for 119895 is a multiple of an

anisotropic (120593 119902 119904)-atom supported in the dilated ball119909119895+ 119861

ℓ119895 with the property

sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf

120582 isin (0infin) sum

119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)

where the infimum is taken over all admissibledecompositions of 119891 as above

Remark 31 (i) In Definition 30 if we assume that 119891 canbe represented as 119891 = sum

119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are

(120593 119902 119904)-atoms supported in dilated balls 119909119895+ 119861

ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)

where the infimum is taken over all admissible decomposi-tions of 119891 as above with

Λ119902(120582

119895119895

)

= inf

120582 isin (0infin)

sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)

then the induced space 120593119902119904119860

(R119899) and the space 119867120593119902119904119860

(R119899)

coincide with equivalent (quasi)normsIndeed if119891 = sum

119895120582119895119886119895inS1015840(R119899) for some (120593 119902 119904)-atoms

119886119895119895 and 120582

119895119895sub C such that Λ

119902(120582

119895) lt infin Write 119886

119895=

120582119895119886119895 It is easy to see that Λ

119902(119886119895) ≲ Λ

119902(120582

119895) lt infin

Conversely if 119891 = sum119895119886119895in S1015840(R119899) with Λ

119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)

we see that 119891 = sum119895120582119895119886119895and Λ

119902(120582

119895) = Λ

119902(119886119895) lt infin Thus

the above claim holds true


(ii) If 120593 is as in (15) with an anisotropic 119860infin(R119899)

Muckenhoupt weight 119908 and Φ(119905) = 119905119901 for all 119905 isin [0infin)

with 119901 isin (0 1] then the atomic space 119867120593119902119904119860

(R119899) is just theweighted anisotropic atomic Hardy space introduced in [6]

The following lemma shows that anisotropic (120593 119902 119904)-atoms of Musielak-Orlicz type are in119867120593

119860(R119899)

Lemma 32 Let (120593 119902 119904) be an anisotropic admissible tripletand let 119898 isin [119904infin) cap Z

+ Then there exists a positive constant

119862 = 119862(120593 119902 119904 119898) such that for any anisotropic (120593 119902 119904)-atom119886 associated with some 119909

0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862

Proof Thecase 119902 = infin is easyWe just consider 119902 isin (119902(120593)infin)Now let us write

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁

sdot sdot sdot = I + II(99)

By using Lemma 10 the proof of I ≲ 120593(1199090+119861

119895 119886

119871119902

120593(1199090+119861119895)) is

similar to that of [20 Lemma 51] the details being omittedTo estimate II we claim that for all ℓ isin Z

+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)

where 119904 ge lfloor(119902(120593)119894(120593) minus 1) ln 119887 ln(120582minus)rfloor If this claim is true

choosing 119902 gt 119902(120593) and 119901 lt 119894(120593) such that 119887minus119902+119901(120582minus)(119904+1)119901

gt 1then by 120593 isin A

119902(119860) and Lemma 10 we have

II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)

Combining the estimates for I and II we obtain (98)To prove the estimate (100) we borrow some techniques

from the proof of Theorem 42 in [9] By Holderrsquos inequality120593 isin A

119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590

) 119896 isin Z and 120601 isin S119904(R119899) For

119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and

119895 + 119896 gt 0 we conclude that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)

For 119895 + 119896 le 0 let 119875 be the Taylor expansion of 120601 at the point119860minus119896

(119909minus1199090) of order 119904Thus by the Taylor remainder theorem

and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)

|119911| for all 119911 isin R119899 (see [9 Section 2])we see that

sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)

where in the last step we used (8) and the fact that

119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)

since ℓ ge 0 By this (103) 119895 + 119896 le 0 and the fact that 119886 hasvanishing moments up to order 119904 we find that1003816100381610038161003816119886 lowast 120601119896 (119909)

1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)

Observe that when 119895+119896+ℓ gt 0 by 119887(120582minus)119904+1

le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)


Finally when 119895+119896+ℓ le 0 from (107) we immediately deduce(108)This shows that (108) holds for all 119895+119896 le 0 Combiningthis with (104) and taking supremum over 119896 isin Z we see that

sup120601isinS119904(R

119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)

From this estimate and 119886lowast119898(119909) ≲ sup

120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|

(see [9 Propostion 310]) we further deduce (100) and hencecomplete the proof of Lemma 37

Then by using Lemma 32 together with an argumentsimilar to that used in the proof of [20 Theorem 51] weobtain the following theorem the details being omitted

Theorem 33 Let (120593 119902 119904) be an admissible triplet and let119898 isin

[119904infin) cap Z+ Then

119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)

and the inclusion is continuous

To obtain the conclusion 119867120593

119898119860(R119899) sub 119867

120593119902119904

119860(R119899)

we use the Calderon-Zygmund decomposition obtained inSection 4 Let 120593 be an anisotropic growth function let 119898 ge

lfloor119902(120593) ln 119887[119894(120593) ln 120582]rfloor and let 119891 isin 119867120593

119898119860(R119899) For each

119896 isin Z as in Definition 19 119891 has a Calderon-Zygmunddecomposition of degree 119904 and height 120582 = 2119896 associated with119891lowast

119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)

Recall that for fixed 119896 isin Z 119909119896119894119894= 119909

119894119894is a sequence in

Ω119896and ℓ119896

119894119894= ℓ

119894119894is a sequence of integers such that (65)

through (69) hold for Ω = Ω119896 120577119896

119894119894= 120577

119894119894are given by

(70) and 119875119896119894119894= 119875

119894119894are projections of 119891 ontoP

119904(R119899) with

respect to the norms given by (71) Moreover for each 119896 isin Z

and 119894 119895 let 119875119896+1119894119895

be the orthogonal projection of (119891 minus 119875119896+1119895

)120577119896

119894

onto P119904(R119899) with respect to the norm associated with 120577119896+1

119895

given by (71) namely the unique element of P119904(R119899) such

that for all 119876 isin P119904(R119899)

intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)

For convenience let 119861119896119894= 119909

119896

119894+ 119861

ℓ119896

119894+120590

Lemmas 34 through 36 are just [9 Lemmas 51 through53] respectively

Lemma 34 The following hold true

(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)


of 119891 such that for all 119894 119895 and 119896 isin Z


10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁

Lemma 36 For every 119896 isin Z sum119894sum119895119875119896+1

119894119895120577119896+1

119895= 0 where the

series converges pointwise and also in S1015840(R119899)

The proof of the following lemma is similar to that of [20Lemma 54] the details being omitted

Lemma 37 Let 119898 isin N and let 119891 isin 119867120593

119898119860(R119899) Then for any

120582 isin (0infin) there exists a positive constant 119862 independent of119891 and 120582 such that

sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)

The following lemma establishes the atomic decomposi-tions for a dense subspace of119867120593

119898119860(R119899)

Lemma 38 Let 119898 ge 119904 ge lfloor119902(120593) ln 119887[119894(120593) ln 120582]rfloor and let 119902 isin

(119902(120593)infin) Then for any 119891 isin 119871119902

120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there

exists a sequence 119886119896119894119896isinZ119894 of multiples of (120593infin 119904)-atoms such

that 119891 = sum119896isinZsum119894 119886

119896

119894converges almost everywhere and also in

S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590

) forall119896 isin Z (117)

(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0

forall119896 isin Z 119894 119895 with 119894 = 119895

(118)

Moreover there exists a positive constant 119862 independent of 119891such that for all 119896 isin Z and 119894

10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)

and for any 120582 isin (0infin)

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) For each 119896 isin Z

119891 has a Calderon-Zygmund decomposition of degree 119904 ge

lfloor119902(120593) ln 119887[119894(120593) ln(120582minus)]rfloor and height 2119896 associated with 119891

lowast

119898

119891 = 119892119896

+ sum119894119887119896

119894as above The conclusions (117) and (118)

follow immediately from (65) and (66) By (76) of Lemma 24and Proposition 6 we know that 119892119896 rarr 119891 in both119867120593

119898119860(R119899)

and S1015840(R119899) as 119896 rarr infin It follows from Lemma 27(ii) that119892119896

119871infin(R119899) rarr 0 as 119896 rarr minusinfin which further implies that

119892119896

rarr 0 almost everywhere as 119896 rarr minusinfin and moreover bythe fact that 119871infin(R119899) is continuously embedding intoS1015840(R119899)(see [6 Lemma 28]) we conclude that 119892119896 rarr 0 in S1015840(R119899) as119896 rarr minusinfin Therefore we obtain

119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)

in S1015840(R119899)Since supp(sum

119894119887119896

119894) sub Ω

119896and 120593(Ω

119896 1) rarr 0 as 119896 rarr infin

then 119892119896

rarr 119891 almost everywhere as 119896 rarr infin Thus (121)also holds almost everywhere By Lemma 36 andsum

119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1

119895for all 119895 we see that

119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1

119895) minus (119891 minussum

119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)

where all the series converge in S1015840(R119899) and almost every-where Furthermore

119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)

By definitions of 119875119896119894and 119875119896+1

119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)

Moreover since sum119895120577119896+1

119895= 120594

Ω119896+1 we rewrite (123) into

119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)

By Lemma 17 we know that |119891(119909)| le 119891lowast119898(119909) le 2

119896+1 for almostevery 119909 isin (Ω

119896+1)∁ and by Lemmas 21 34(ii) and 35 we find

that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896

119894= 0 and hence by

Lemma 34(i) we see that supp 120577119896+1119895

sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

Therefore by applying (123) we further conclude that

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)

Obviously (126) and (127) imply (119) and (116) respec-tively Moreover by (124) (126) and (127) we know that 119886119896

119894

is a multiple of a (120593infin 119904)-atom By Lemma 10 (118) (126)uniformly upper type 1 property of 120593 and Lemma 37 for any120582 isin (0infin) we have

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)

which gives (120) This finishes the proof of Lemma 38

The following Lemma 39 is just [20 Lemma 43(ii)]

Lemma 39 Let 120593 be an anisotropic growth function For anygiven positive constant 119888 there exists a positive constant119862 suchthat for some 120582 isin (0infin) the inequalitysum

119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)

Theorem 40 Let 119902(120593) be as in (13) If 119898 ge 119904 ge

lfloor119902(120593) ln 119887[119894(120593) ln 120582]rfloor and 119902 isin (119902(120593)infin] then 119867120593119902119904119860

(R119899) =

119867120593

119898119860(R119899) = 119867

120593

119860(R119899) with equivalent (quasi)norms

Proof Observe that by (103) Definition 30 andTheorem 33it holds true that

119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)

where 119898 ge 119904 ge lfloor119902(120593) ln 119887[119894(120593) ln 120582]rfloor and all the inclusionsare continuous Thus to finish the proof of Theorem 40 itsuffices to prove that for all 119891 isin 119867

120593

119898119860(R119899) with 119898 ge

119904 ge lfloor119902(120593) ln 119887[119894(120593) ln 120582]rfloor 119891119867120593infin119904

119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)

To this end let 119891 isin 119867120593

119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)


Consequently by Lemma 39 we see that

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593

119898119860(R119899) By Corollary 28 there exists a

sequence 119891119896119896isinN of functions in119867120593

119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593

119898119860(R119899) By Lemma 38 for each 119896 isin N 119891

119896has an atomic

decomposition 119891119896= sum

119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are

multiples of (120593infin 119904)-atoms with supp 119889119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)

then by Lemma 39 we further see that 119891 = sum119896isinNsum119894isinN 119889

119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)

which completes the proof of Theorem 40

For simplicity fromnow on we denote simply by119867120593119860(R119899)

the anisotropic Hardy space 119867120593119898119860

(R119899) of Musielak-Orlicztype with119898 ge 119898(120593)

6 Finite Atomic Decompositions andTheir Applications

The goal of this section is to obtain the finite atomic decom-position characterization of119867120593

119860(R119899) and as an application a

bounded criterion on119867120593119860(R119899) of quasi-Banach space-valued

sublinear operators is also obtained

61 Finite Atomic Decompositions In this subsection weprove that for any given finite linear combination of atomswhen 119902 lt infin (or continuous atoms when 119902 = infin) itsnorm in 119867

120593

119860(R119899) can be achieved via all its finite atomic

decompositions This extends the conclusion [39 Theorem31] by Meda et al to the setting of anisotropic Hardy spacesof Musielak-Orlicz type

Definition 41 Let (120593 119902 119904) be an admissible triplet Denote by119867120593119902119904

119860fin(R119899

) the set of all finite linear combinations of multiples

of (120593 119902 119904)-atoms and the norm of 119891 in119867120593119902119904119860fin(R

119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)

Obviously for any admissible triplet (120593 119902 119904) the set119867120593119902119904

119860fin(R119899

) is dense in 119867120593119902119904

119860(R119899) with respect to the quasi-

norm sdot 119867120593119902119904

119860(R119899)

In order to obtain the finite atomic decomposition weneed the notion of the uniformly locally dominated conver-gence condition from [20] An anisotropic growth function 120593is said to satisfy the uniformly locally dominated convergencecondition if the following holds for any compact set 119870 in R119899

and any sequence 119891119898119898isinN of measurable functions such that

119891119898(119909) tends to 119891(119909) for almost every 119909 isin R119899 if there exists a

nonnegative measurable function 119892 such that |119891119898(119909)| le 119892(119909)

for almost every 119909 isin R119899 and


int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0

as 119898 997888rarr infin

(137)

We remark that the anisotropic growth functions 120593(119909 119905) =119905119901

[log(119890 + |119909|) + log(119890 + 119905119901)]119901 for all 119909 isin R119899 and 119905 isin (0infin)

with 119901 isin (0 1) and 120593 as in (15) satisfy the uniformly locallydominated convergence condition see [20 page 12]

Theorem 42 Let 120593 be an anisotropic growth function satis-fying the uniformly locally dominated convergence condition119902(120593) as in (13) and (120593 119902 119904) an admissible triplet

(i) If 119902 isin (119902(120593)infin) then sdot 119867120593119902119904

119860fin (R119899)and sdot

119867120593

119860(R119899) are

equivalent quasinorms on119867120593119902119904119860fin(R

119899

)

(ii) sdot 119867120593infin119904


119867120593

119860(R119899) are equivalent quasinorms

on119867120593infin119904119860fin (R119899) cap C(R119899)

Proof Obviously by Theorem 40119867120593119902119904119860fin(R

119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)

Thus we only need to prove that for all 119891 isin 119867120593119902119904

119860fin(R119899

) when119902 isin (119902(120593)infin) and for all119891 isin 119867

120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)


Now we prove this by three steps

Step 1 (a new decomposition of119891 isin 119867120593119902119904

119860119891119894119899(R119899)) Assume that

119902 isin (119902(120593)infin]Without loss of generality wemay assume that119891 isin 119867

120593119902119904

119860fin(R119899

) and 119891119867120593

119860(R119899) = 1 Notice that 119891 has compact

support Suppose that supp119891 sub 119861 = 1198611198960for some 119896

0isin Z

where 1198611198960is as in Section 2 For each 119896 isin Z let

Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)

We use the same notation as in Lemma 38 Since 119891 isin

119867120593

119860(R119899) cap 119871

119902

120593(sdot1)(R119899) where 119902 = 119902 if 119902 isin (119902(120593)infin) and

119902 = 119902(120593) + 1 if 119902 = infin by Lemma 38 there exists asequence 119886119896

119894119896isinZ119894

of multiples of (120593infin 119904)-atoms such that119891 = sum

119896isinZsum119894 119886119896

119894holds almost everywhere and in S1015840(R119899)

Moreover by 119867120593infin119904119860

(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)

On the other hand by Step 2 of the proof of [6 Theorem62] we know that there exists a positive constant 119862 depend-ing only on 119898(120593) such that 119891lowast(119909) le 119862 inf

119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590

)∁ Hence for all 119909 isin (119861lowast)∁ we have

119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)

We now denote by 1198961015840 the largest integer 119896 such that 2119896 lt119862120594

119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896

119894and let ℓ = sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where

the series converge almost everywhere and inS1015840(R119899) Clearly119891 = ℎ + ℓ and supp ℓ sub ⋃

119896gt1198961015840 Ω119896sub 119861

lowast which together withsupp119891 sub 119861

lowast further yields supp ℎ sub 119861lowast

Step 2 (prove ℎ to be a multiple of a (120593 119902 119904)-atom) Noticethat for any 119902 isin (119902(120593)infin] and 119902

1isin (1 119902119902(120593)) by Holderrsquos

inequality and 120593 isin A1199021199021

(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)

Observing that supp119891 sub 119861 and 119891 has vanishing moments upto order 119904 we know that 119891 is a multiple of a (1 119902

1 0)-atom

and therefore 119891lowast isin 1198711

(R119899) Then by (142) (116) (117) and(119) of Lemmas 38 and 34(ii) for any |120572| le 119904 we concludethat

intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)

(Notice that 119886119896119894in (119) is replaced by 120582

119896

119894119886119896

119894here) This

together with the vanishingmoments of 119886119896119894 implies that ℓ has

vanishing moments up to order 119904 and hence so does ℎ byℎ = 119891 minus ℓ Using Lemma 34(ii) (119) of Lemma 38 and thefacts that 2119896

1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)

Thus there exists a positive constant 1198620 independent of 119891

such that ℎ1198620is a (120593infin 119904)-atom and by Definition 30 it is

also a (120593 119902 119904)-atom for any admissible triplet (120593 119902 119904)

Step 3 (prove (i)) Let 119902 isin (119902(120593)infin) We first showsum119896gt1198961015840 sum119894120582119896

119894119886119896

119894isin 119871

119902

120593(sdot1)(R119899) For any 119909 isin R119899 since R119899 =

cup119896isinZ(Ω119896 Ω119896+1) there exists 119895 isin Z such that 119909 isin (Ω

119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying

Lemma 34(ii) and (119) of Lemma 38 we conclude that forall 119909 isin (Ω

119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast

) we further have 119891lowast isin 119871119902

120593(119861lowast

)Since120593 satisfies the uniformly locally dominated convergencecondition it follows that sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges to ℓ in

119871119902

120593(119861lowast

)Now for any positive integer 119870 let

119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)

and let ℓ119870= sum

(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast

) for any 120598 isin (0 1) if 119870 is large enough we have that(ℓ minus ℓ

119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a

finite linear combination of atoms By (120) of Lemma 38 andStep 2 we further find that

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)

which completes the proof of (i)To prove (ii) assume that 119891 is a continuous function

in 119867120593infin119904

119860fin (R119899

) then 119886119896

119894is also continuous by examining its

definition (see (123)) Since 119891lowast(119909) le 119862119899119898(120593)

119891119871infin(R119899) for any

119909 isin R119899 where the positive constant 119862119899119898(120593)

only depends on119899 and 119898(120593) it follows that the level set Ω

119896is empty for all 119896

satisfying that 2119896 ge 119862119899119898(120593)

119891119871infin(R119899) We denote by 11989610158401015840 the

largest integer for which the above inequality does not holdThen the index 119896 in the sum defining ℓ will run only over1198961015840

lt 119896 le 11989610158401015840

Let 120598 isin (0infin) Since119891 is uniformly continuous it followsthat there exists a 120575 isin (0infin) such that if 120588(119909 minus 119910) lt 120575 then|119891(119909) minus 119891(119910)| lt 120598 Write ℓ = ℓ

120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840


On the other hand for any fixed integer 119896 isin (1198961015840

11989610158401015840

] by(118) of Lemma 38 and Ω

119896sub 119861

lowast we see that 1198651is a finite set

and hence ℓ1205981is continuous Furthermore from Step 5 of the

proof of [6 Theorem 62] it follows that |ℓ1205982| le 120598(119896

10158401015840

minus 1198961015840

)Since 120598 is arbitrary we can hence split ℓ into a continuouspart and a part that is uniformly arbitrarily small This factimplies that ℓ is continuousThus ℎ = 119891minus ℓ is a multiple of acontinuous (120593infin 119904)-atom by Step 2

Now we can give a finite atomic decomposition of 119891 Letus use again the splitting ℓ = ℓ

120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is

a finite sum of multiples of (120593infin 119904)-atoms and1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)

Notice that ℓ ℓ1205981are continuous and have vanishing moments

up to order 119904 and hence so does ℓ1205982= ℓ minus ℓ

120598

1 Moreover

supp ℓ1205982sub 119861

lowast and ℓ1205982119871infin(R119899)

le 1198621(11989610158401015840

minus 1198961015840

)120598 Thus we canchoose 120598 small enough such that ℓ120598

2becomes a sufficient small

multiple of a continuous (120593infin 119904)-atom that is

ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598

2is a finite linear combination of

continuous atoms Then by (150) and the fact that ℎ1198620is a

(120593infin 119904)-atom we have10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)

This finishes the proof of (ii) and henceTheorem 42

62 Applications As an application of the finite atomicdecompositions obtained in Theorem 42 we establish theboundedness on 119867

120593

119860(R119899) of quasi-Banach-valued sublinear

operatorsRecall that a quasi-Banach space B is a vector space

endowed with a quasinorm sdot B which is nonnegativenondegenerate (ie 119891B = 0 if and only if 119891 = 0) andhomogeneous and obeys the quasitriangle inequality that isthere exists a constant 119870 isin [1infin) such that for all 119891 119892 isin B119891 + 119892B le 119870(119891B + 119892B)

Definition 43 Let 120574 isin (0 1] A quasi-Banach space B120574

with the quasinorm sdot B120574is a 120574-quasi-Banach space if

119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574

Notice that any Banach space is a 1-quasi-Banach spaceand the quasi-Banach spaces ℓ119901 119871119901

119908(R119899) and 119867

119901

119908(R119899 119860)

with 119901 isin (0 1] are typical 119901-quasi-Banach spaces Alsowhen 120593 is of uniformly lower type 119901 isin (0 1] the space119867120593

119860(R119899) is a 119901-quasi-Banach space Moreover according to

the Aoki-Rolewicz theorem (see [47] or [48]) any quasi-Banach space is in essential a 120574-quasi-Banach space where120574 = log

2(2119870)

minus1

For any given 120574-quasi-Banach space B120574with 120574 isin (0 1]

and a sublinear spaceY an operator 119879 fromY toB120574is said

to beB120574-sublinear if for any 119891 119892 isin Y and complex numbers

120582 ] it holds true that1003817100381710038171003817119879(120582119891 + ]119892)1003817100381710038171003817

120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)

and 119879(119891) minus 119879(119892)B120574 le 119879(119891 minus 119892)B120574 We remark that if 119879 is linear then 119879 is B

120574-sublinear

Moreover if B120574

= 119871119902

119908(R119899) with 119902 isin [1infin] 119908 is a

Muckenhoupt 119860infin

weight and 119879 is sublinear in the classicalsense then 119879 is alsoB

120574-sublinear

Theorem 44 Let (120593 119902 119904) be an admissible triplet Assumethat 120593 is an anisotropic growth function satisfying the uni-formly locally dominated convergence condition and being ofuniformly upper type 120574 isin (0 1] andB

120574a quasi-Banach space

If one of the following holds true(i) 119902 isin (119902(120593)infin) and 119879 119867

120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-

sublinear operator such that

119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)

(ii) 119879 is a B120574-sublinear operator defined on continuous

(120593infin 119904)-atoms such that

119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)

then 119879 has a unique bounded B120574-sublinear operator

extension from119867120593

119860(R119899) toB

120574

Proof Suppose that the assumption (i) holds true For any119891 isin 119867

120593119902119904

119860fin(R119899

) by Theorem 42(i) there exist complexnumbers 120582

119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By

Remark 31(i) we know that

Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)

Since120593 is of uniformly upper type 120574 it follows that there existsa positive constant 119862

120574such that for all 119909 isin R119899 119904 isin [1infin)

and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)


If there exists 1198950isin 1 119896 such that 119862

120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)

Otherwise it follows from (157) that

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)

Therefore by the assumption (i) we obtain

10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899

) is dense in119867120593119860(R119899) a density argument then

gives the desired conclusionSuppose now that the assumption (ii) holds true Similar

to the proof of (i) by Theorem 42(ii) we also conclude thatfor all 119891 isin 119867

120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To

extend 119879 to the whole 119867120593119860(R119899) we only need to prove that

119867120593infin119904

119860fin (R119899

)capC(R119899) is dense in119867120593119860(R119899) Since119867119901infin119904

120593fin (R119899 119860)

is dense in119867120593119860(R119899) it suffices to prove119867120593infin119904

119860fin (R119899

) capC(R119899) isdense in119867119901infin119904

120593fin (R119899 119860) in the quasinorm sdot 119867120593

119860(R119899) Actually

we only need to show that119867120593infin119904119860fin (R

119899

) cap Cinfin(R119899) is dense in119867120593infin119904

119860fin (R119899

) due toTheorem 42To see this let 119891 isin 119867

120593infin119904

119860fin (R119899

) Since 119891 is a finite linearcombination of functions with bounded supports it followsthat there exists 119897 isin Z such that supp119891 sub 119861

119897 Take 120595 isin S(R119899)

such that supp120595 sub 1198610and int

R119899120595(119909) 119889119909 = 1 By (7) it is easy

to show that supp(120595119896lowast119891) sub 119861

119897+120590for anyminus119896 lt 119897 and119891lowast120595

119896has

vanishing moments up to order 119904 where 120595119896(119909) = 119887

119896

120595(119860119896

119909)

for all 119909 isin R119899 Hence 119891 lowast 120595119896isin 119867

120593infin119904

119860fin (R119899

) cap Cinfin(R119899)Likewise supp(119891 minus 119891 lowast 120595

119896) sub 119861

119897+120590for any minus119896 lt 119897

and 119891 minus 119891 lowast 120595119896has vanishing moments up to order 119904 Take

any 119902 isin (119902(120593)infin) By [6 Proposition 29 (ii)] and the fact

that 120593 satisfies the uniformly locally dominated convergencecondition we know that

1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902

120593(119861119897+120590)997888rarr 0 as 119896 997888rarr infin (162)

and hence 119891minus119891lowast120595119896= 119888119896119886119896for some (120593 119902 119904)-atom 119886

119896 where

119888119896is a constant depending on 119896 and 119888

119896rarr 0 as 119896 rarr infinThus

we obtain 119891 minus 119891 lowast 120595119896119867120593

119860(R119899) rarr 0 as 119896 rarr infin This finishes

the proof of Theorem 44

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundation of China (Grants nos 11001234 1116104411171027 11361020 and 11101038) the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina (Grant no 20120003110003) and the FundamentalResearch Funds for Central Universities of China (Grant no2012LYB26)

References

[1] C Fefferman and E M Stein ldquo119867119901 spaces of several variablesrdquoActa Mathematica vol 129 no 3-4 pp 137ndash193 1972

[2] G B Folland and E M Stein Hardy Spaces on HomogeneousGroups vol 28 of Mathematical Notes Princeton UniversityPress Princeton NJ USA 1982

[3] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009

[4] S Muller ldquoHardy space methods for nonlinear partial differen-tial equationsrdquo TatraMountainsMathematical Publications vol4 pp 159ndash168 1994

[5] J-O Stromberg and A TorchinskyWeighted Hardy Spaces vol1381 of LectureNotes inMathematics Springer Berlin Germany1989

[6] M Bownik B Li D Yang and Y Zhou ldquoWeighted anisotropicHardy spaces and their applications in boundedness of sublin-ear operatorsrdquo Indiana University Mathematics Journal vol 57no 7 pp 3065ndash3100 2008

[7] A-P Calderon and A Torchinsky ldquoParabolic maximal func-tions associated with a distribution IIrdquo Advances in Mathemat-ics vol 24 no 2 pp 101ndash171 1977

[8] J Garcıa-Cuerva ldquoWeighted119867119901 spacesrdquo Dissertationes Mathe-maticae vol 162 pp 1ndash63 1979

[9] M Bownik ldquoAnisotropic Hardy spaces and waveletsrdquoMemoirsof theAmericanMathematical Society vol 164 no 781 p vi+1222003

[10] M Bownik ldquoOn a problem of Daubechiesrdquo ConstructiveApproximation vol 19 no 2 pp 179ndash190 2003

[11] Z Birnbaum and W Orlicz ldquoUber die verallgemeinerungdes begriffes der zueinander konjugierten potenzenrdquo StudiaMathematica vol 3 pp 1ndash67 1931


[12] W Orlicz ldquoUber eine gewisse Klasse von Raumen vom TypusBrdquo Bulletin International de lrsquoAcademie Polonaise des Sciences etdes Lettres Serie A vol 8 pp 207ndash220 1932

[13] K Astala T Iwaniec P Koskela and G Martin ldquoMappingsof BMO-bounded distortionrdquoMathematische Annalen vol 317no 4 pp 703ndash726 2000

[14] T Iwaniec and J Onninen ldquoH1-estimates of Jacobians bysubdeterminantsrdquo Mathematische Annalen vol 324 no 2 pp341ndash358 2002

[15] S Martınez and N Wolanski ldquoA minimum problem with freeboundary in Orlicz spacesrdquo Advances in Mathematics vol 218no 6 pp 1914ndash1971 2008

[16] J-O Stromberg ldquoBounded mean oscillation with Orlicz normsand duality of Hardy spacesrdquo Indiana University MathematicsJournal vol 28 no 3 pp 511ndash544 1979

[17] S Janson ldquoGeneralizations of Lipschitz spaces and an appli-cation to Hardy spaces and bounded mean oscillationrdquo DukeMathematical Journal vol 47 no 4 pp 959ndash982 1980

[18] R Jiang and D Yang ldquoNew Orlicz-Hardy spaces associatedwith divergence form elliptic operatorsrdquo Journal of FunctionalAnalysis vol 258 no 4 pp 1167ndash1224 2010

[19] J Garcıa-Cuerva and J MMartell ldquoWavelet characterization ofweighted spacesrdquoThe Journal of Geometric Analysis vol 11 no2 pp 241ndash264 2001

[20] L D Ky ldquoNew Hardy spaces of Musielak-Orlicz type andboundedness of sublinear operatorsrdquo Integral Equations andOperator Theory vol 78 no 1 pp 115ndash150 2014

[21] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer Berlin Germany 1983

[22] Y Liang J Huang and D Yang ldquoNew real-variable characteri-zations ofMusielak-Orlicz Hardy spacesrdquo Journal of Mathemat-ical Analysis and Applications vol 395 no 1 pp 413ndash428 2012

[23] L Diening ldquoMaximal function on Musielak-Orlicz spacesand generalized Lebesgue spacesrdquo Bulletin des SciencesMathematiques vol 129 no 8 pp 657ndash700 2005

[24] A Bonami and S Grellier ldquoHankel operators and weak factor-ization for Hardy-Orlicz spacesrdquo Colloquium Mathematicumvol 118 no 1 pp 107ndash132 2010

[25] A Bonami S Grellier and LD Ky ldquoParaproducts and productsof functions in119861119872119874(119877

119899

) and1198671(119877119899) throughwaveletsrdquo JournaldeMathematiques Pures et Appliquees vol 97 no 3 pp 230ndash2412012

[26] A Bonami T Iwaniec P Jones and M Zinsmeister ldquoOn theproduct of functions in BMO and 119867

1rdquo Annales de lrsquoInstitutFourier vol 57 no 5 pp 1405ndash1439 2007

[27] L Diening P Hasto and S Roudenko ldquoFunction spaces ofvariable smoothness and integrabilityrdquo Journal of FunctionalAnalysis vol 256 no 6 pp 1731ndash1768 2009

[28] E Nakai and K Yabuta ldquoPointwise multipliers for functions ofbounded mean oscillationrdquo Journal of the Mathematical Societyof Japan vol 37 no 2 pp 207ndash218 1985

[29] E Nakai and K Yabuta ldquoPointwise multipliers for functions ofweighted bounded mean oscillation on spaces of homogeneoustyperdquoMathematica Japonica vol 46 no 1 pp 15ndash28 1997

[30] L D Ky ldquoBilinear decompositions for the product space 1198671119871times

119861119872119874119871rdquoMathematische Nachrichten 2013

[31] A Bonami J Feuto and S Grellier ldquoEndpoint for the DIV-CURL lemma inHardy spacesrdquo PublicacionsMatematiques vol54 no 2 pp 341ndash358 2010

[32] L D Ky ldquoBilinear decompositions and commutators of singularintegral operatorsrdquo Transactions of the American MathematicalSociety vol 365 no 6 pp 2931ndash2958 2013

[33] R R Coifman ldquoA real variable characterization of 119867119901rdquo StudiaMathematica vol 51 pp 269ndash274 1974

[34] R H Latter ldquoA characterization of 119867119901(R119899) in terms of atomsrdquoStudia Mathematica vol 62 no 1 pp 93ndash101 1978

[35] Y Meyer M H Taibleson and G Weiss ldquoSome functionalanalytic properties of the spaces 119861

119902generated by blocksrdquo

Indiana University Mathematics Journal vol 34 no 3 pp 493ndash515 1985

[36] M Bownik ldquoBoundedness of operators on Hardy spaces viaatomic decompositionsrdquo Proceedings of the American Mathe-matical Society vol 133 no 12 pp 3535ndash3542 2005

[37] Y Meyer and R Coifman Wavelet Calderon-Zygmund andMultilinear Operators Cambridge Studies in Advanced Mathe-matics 48 CambridgeUniversity Press CambridgeMass USA1997

[38] D Yang and Y Zhou ldquoA boundedness criterion via atoms forlinear operators in Hardy spacesrdquo Constructive Approximationvol 29 no 2 pp 207ndash218 2009

[39] S Meda P Sjogren and M Vallarino ldquoOn the 1198671-1198711 bound-edness of operatorsrdquo Proceedings of the American MathematicalSociety vol 136 no 8 pp 2921ndash2931 2008

[40] M Bownik B Li D Yang and Y Zhou ldquoWeighted anisotropicproduct Hardy spaces and boundedness of sublinear operatorsrdquoMathematische Nachrichten vol 283 no 3 pp 392ndash442 2010

[41] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mathematics Springer Berlin Germany 1971

[42] M Bownik and K-P Ho ldquoAtomic and molecular decomposi-tions of anisotropic Triebel-Lizorkin spacesrdquoTransactions of theAmerican Mathematical Society vol 358 no 4 pp 1469ndash15102006

[43] R Johnson and C J Neugebauer ldquoHomeomorphisms preserv-ing 119860

119901rdquo Revista Matematica Iberoamericana vol 3 no 2 pp

249ndash273 1987[44] S Janson ldquoOn functions with conditions on the mean oscilla-

tionrdquo Arkiv for Matematik vol 14 no 2 pp 189ndash196 1976[45] B Muckenhoupt and R L Wheeden ldquoOn the dual of weighted

1198671 of the half-spacerdquo Studia Mathematica vol 63 no 1 pp 57ndash

79 1978[46] H Q Bui ldquoWeighted Hardy spacesrdquo Mathematische Nachrich-

ten vol 103 pp 45ndash62 1981[47] T Aoki ldquoLocally bounded linear topological spacesrdquo Proceed-

ings of the Imperial Academy vol 18 no 10 pp 588ndash594 1942[48] S Rolewicz Metric Linear Spaces PWNmdashPolish Scientific

Publishers Warsaw Poland 2nd edition 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


eg [20ndash23]) Recently Ky [20] introduced a new Musielak-Orlicz Hardy space119867120593(R119899) via the grand maximal functionand established its atomic characterization It is knownthat 119867120593(R119899) generalizes both the Orlicz-Hardy space ofStromberg [16] and Janson [17] and the weightedHardy space119867119901

119908(R119899)with119908 isin A

infin(R119899) studied by Garcıa-Cuerva [8] and

Stromberg and Torchinsky [5] Recall that the motivation tostudy function spaces of Musielak-Orlicz type comes fromtheir applications to many branches of mathematics andphysics (see eg [20 23ndash27]) In [20] Ky further introducedthe BMO-type space BMO

120593(R119899) which was proven to be

the dual space of 119867120593(R119899) as an interesting application Kyproved that the class of pointwise multipliers for BMO(R119899)characterized by Nakai and Yabuta [28 29] is the dual spaceof 1198711(R119899) +119867log

(R119899) where119867log(R119899) denotes the Musielak-

Orlicz Hardy space related to the growth function

120593 (119909 119905) =119905

log (119890 + |119909|) + log (119890 + 119905) (1)

for all 119909 isin R119899 and 119905 isin [0infin) It is worth noticing that somespecial Musielak-Orlicz Hardy spaces appear naturally in thestudy of the products of functions in BMO(R119899) and 1198671(R119899)(see [25 26 30]) the endpoint estimates for the div-curllemma and the commutators of singular integral operators(see [25 30ndash32])

Moreover observe that a distribution inHardy spaces canbe represented as a (finite or infinite) linear combination ofatoms (see [33 34]) Then the boundedness of linear oper-ators in Hardy spaces can be deduced from their behavioron atoms in principle However Meyer et al [35 page 513]gave an example of 119891 isin 119867

1

(R119899) whose norm can not beachieved by its finite atomic decompositions via (1infin 0)-atoms Applying this Bownik [36] showed that there existsa linear functional defined on a dense subspace of 1198671(R119899)which maps all (1infin 0)-atoms into bounded scalars butyet can not extend to a bounded linear functional on thewhole 1198671(R119899) Let 119901 isin (0 1] and let 119904 be a nonnegativeinteger not less than 119899(1119901 minus 1) This implies that theuniform boundedness in some quasi-Banach space B of alinear operator 119879 on all (119901infin 119904)-atoms does not generallyguarantee the boundedness of 119879 from 119867

119901

(R119899) to B Thisphenomenon has also essentially already been observed byMeyer et al in [37 page 19] Motivated by [36] via usingthe Lusin function characterization of Hardy spaces119867119901(R119899)Yang and Zhou [38] proved that a B

120574-sublinear operator

119879 uniquely extends to a bounded B120574-sublinear operator

from 119867119901

(R119899) with 119901 isin (0 1] to some quasi-Banach spaceB if and only if 119879 maps all (119901 2 119904)-atoms into uniformlybounded elements of B Independently Meda et al [39]established anothermore general bounded criterion via usingthe grand maximal function characterization of 119867119901(R119899)precisely they proved that if 119879 is a linear operator and mapsall (119901 119902 119904)-atoms with 119902 lt infin or all continuous (119901infin 119904)-atoms into uniformly bounded elements of a Banach spaceB then 119879 uniquely extends to a bounded linear operatorfrom 119867

119901

(R119899) to B This result was further generalizedto the weighted anisotropic Hardy spaces in [6] weighted

anisotropic product Hardy spaces in [40] and especiallyHardy spaces of Musielak-Orlicz type by Ky in [20]

There are three goals in this paper First we introduceanisotropic Hardy spaces of Musielak-Orlicz type 119867120593

119860(R119899)

via grand maximal functions and characterize these spacesvia anisotropic atomic decompositions These Hardy spacesinclude classical Hardy spaces119867119901(R119899) of Fefferman and Stein[1] weighted anisotropic Hardy spaces of Bownik [6] andHardy spaces of Musielak-Orlicz type of Ky [20]

The second goal is to obtain some new real-variable char-acterizations of119867120593

119860(R119899) in terms of the radial the nontangen-

tial and the tangential maximal functions via some boundedestimates of the truncated maximal function pointwise orin anisotropic Musielak-Orlicz spaces which are motivatedby [9 Section 7] These real-variable characterizations of119867120593

119860(R119899) coincide with the known best results when119867120593

119860(R119899)

is the anisotropic Hardy space 119867119901119860(R119899) with 119901 isin (0 1] (see

[9 Theorem 71]) or new even in its weighted variantThe third goal is to generalize the result ofMeda et al [39]

to the present setting More precisely we prove the existenceof finite atomic decompositions achieving the norm in densesubspaces of 119867120593

119860(R119899) As an application we prove that for a

given admissible triplet (120593 119902 119904) (seeDefinition 30 below) if119879is a B

120574-sublinear operator and maps all (120593 119902 119904)-atoms with

119902 lt infin (or all continuous (120593 119902 119904)-atoms with 119902 = infin) intouniformly bounded elements of some quasi-Banach spacesB then 119879 uniquely extends to a bounded B

120574-sublinear

operator from 119867120593

119860(R119899) to B These results are new even for

the anisotropic Hardy-Orlicz spaces on R119899This paper is organized as follows In Section 2 we first

recall some notation and definitions concerning Musielak-Orlicz functions expansive dilations and Muckenhouptweights Then we introduce the anisotropic Hardy spaces ofMusielak-Orlicz type119867120593

119860(R119899) via grand maximal functions

and some basic properties of these spaces are also presentedIn Section 3 we obtain some new real-variable characteri-zations of 119867120593

119860(R119899) via the radial the nontangential and the

tangential maximal functions Section 4 is devoted to gener-alizing the Calderon-Zygmund decomposition associated toweighted anisotropic Hardy spaces in [6] to the more generalspaces119867120593

119860(R119899) Applying this in Section 5 we introduce the

anisotropic atomic Hardy spaces of Musielak-Orlicz type119867120593119902119904

119860(R119899) for any admissible triplet (120593 119902 119904) and further

prove that for any admissible triplet (120593 119902 119904)

119867120593

119860(R119899

) = 119867120593119902119904

119860(R119899

) (2)

with equivalent norms (see Theorem 40 below) Moreoverin Section 61 we prove that sdot

119867120593119902119904

119860fin(R119899)and sdot

119867120593

119860(R119899) are

equivalent quasinorms on 119867120593119902119904

119860fin(R119899

) when 119902 lt infin and on119867120593119902119904

119860fin(R119899

) cap C(R119899) when 119902 = infin where 119867120593119902119904119860fin(R

119899

) denotesthe space of all finite linear combinations of multiples of(120593 119902 119904)-atoms In Section 62 we obtain criteria for bound-edness of sublinear operators in 119867

120593

119860(R119899) (see Theorem 44

below) The results in Section 6 are also new even for theanisotropic Hardy-Orlicz spaces on R119899

Finally we make some conventions on notation Let N =

1 2 and let Z+= 0 cup N Denote by S(R119899) the space



1 120572

119899) isin Z119899

+ |120572| = 120572

1+


120572

= (1205971205971199091)1205721 sdot sdot sdot (120597120597119909

119899)120572119899 Throughout the











120601 (119904119905) le 119862119904119901

120601 (119905) (3)


120593 (119909 119904119905) le 119862119904119901

120593 (119909 119905) (4)


(0infin) Let





(5)





+be two positive

numbers such that


+

(6)


+= max|120582| 120582 isin 120590(119860)



119896= 119860

119896


119896sub 119903119861

119896sub

119861119896+1

and |119861119896| = 119887


119861119896+ 119861

119895sub 119861

119895+120590 (7)

119861119896+ (119861

119896+120590)∁

sub (119861119896)∁

(8)



119860 R119899 rarr [0infin)









120588 (119909) = sum

119896isinZ

119887119896

120594119861119896+1119861119896

(119909) if 119909 = 0 or else 120588 (0) = 0 (9)


120588 (119909 + 119910) le 119887120590

(max 120588 (119909) 120588 (119910)) le 119887120590 [120588 (119909) + 120588 (119910)] (10)




119901(119860) denoted by 120593 isin A



(1infin)


sup119896isinZ

119887minus119896

int119909+119861119896

120593 (119910 119905) 119889119910


int119909+119861119896

[120593(119910 119905)]minus1(119901minus1)

119889119910

119901minus1

le 119862

(11)


and when 119901 = 1


sup119896isinZ

119887minus119896

int119909+119861119896

120593 (119910 119905) 119889119910ess sup119910isin119909+119861119896

[120593(119910 119905)]minus1

le 119862

(12)


(120593)Define A

infin(119860) = ⋃

1le119901ltinfinA119901(119860) and





119902(120593)(119860)


A119902(119860)) A

1(119860) such










119898(120593) = lfloor(119902 (120593)

119894 (120593)minus 1)

ln 119887ln 120582

minus

rfloor (14)

Clearly

120593 (119909 119905) = 119908 (119909)Φ (119905) forall119909 isin R119899

119905 isin [0infin) (15)






intR119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 lt infin (16)



R119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 le 1

(17)


S119898(R119899

) = 120601 isin S (R119899

)

sup119909isinR119899|120572|le119898+1

[1 + 120588(119909)]119898+2 1003816100381610038161003816120597

120572

120601 (119909)1003816100381610038161003816 le 1

(18)


119896

120593(119860119896



119891lowast

119898(119909) = sup

120601isinS119898(R119899)

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816119891 lowast 120601119896 (119910)1003816100381610038161003816 (19)





119898isin 119871

120593


119898119860(R119899) = 119891

lowast

119898119871120593(R119899)

When119898 = 119898(120593)119867120593


119860(R119899)










(R119899)


(R119899) sub


Proof Let 119891 isin 119867120593


0


119909(119910) = 120601(119909 minus 119910) for all

119910 isin R119899



sup119909isin1198610 119910isinR

119899

1 + 120588 (119910)

1 + 120588 (119909 minus 119910)le 119887

2120590

(20)


1003816100381610038161003816⟨119891 120601⟩1003816100381610038161003816 =

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

1003816100381610038161003816100381610038161003816100381610038161003816

119891 lowast (120595119909

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

) (119909)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)inf119909isin1198610

119891lowast

119898(119909)

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)

100381710038171003817100381710038171205941198610

10038171003817100381710038171003817

minus1

119871120593(R119899)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(21)




119898119860(R119899) is complete


Maximal Functions






M0

120595119891 (119909) = sup

119896isinZ

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

M120595119891 (119909) = sup

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119879119873

120595119891 (119909) = sup

119896isinZ


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816


+

(22)




119891 isin 119867120593

119860(R119899

) (23)

119879119873

120595119891 isin 119871

120593

(R119899

) 119873 gt[119902 (120593)]

2

119894 (120593) (24)

M120595119891 isin 119871

120593

(R119899

) (25)

M0

120595119891 isin 119871

120593

(R119899

) (26)



120593

119860(R119899) such that

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

=1003817100381710038171003817119891lowast

119898

1003817100381710038171003817119871120593(R119899) le 119862110038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198622

10038171003817100381710038171003817M0

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198623

10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198624

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(27)



120593 (119864 119905) = int119864

120593 (119909 119905) 119889119909 (28)



(119909 + 119861119896) and 119905 isin (0infin)

120593 (119909 + 119861119896 119905)

120593 (119864 119905)le 119862

1003816100381610038161003816119909 + 1198611198961003816100381610038161003816119902

|119864|119902

(29)


119905

0



119898119870119871

(119909 119896) = [max 1 120588 (119860minus119870119909)]119871

(1 + 119887minus119896minus119870

)119871 (30)


M0(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

119898119870119871

(119909 119896)

M(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)

119879119873(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

[1 + 120588 (119860minus119896 (119909 minus 119910))]119873

119898119870119871

(119910 119896)

119873 isin Z+

1198910lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M0(119870119871)

120595119891 (119909)

119891lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M(119870119871)

120595119891 (119909)

(31)

where S119898(R119899) is as in (18)




119865lowast119870

119895(119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119895+119896

119865 (119910 119896) (32)



lowast119870

119895(119909) gt 120582 is open for any


119895which




120593 (119909 isin R119899 119865lowast119870119895

(119909) gt 119905 120582)

le 1198621198871199022119895

120593 (119909 isin R119899 119865lowast1198700

(119909) gt 119905 120582)

(33)

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 le 119862119887

1199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909 (34)



119895(119909) gt 119905 there exist 119896 le 119870

and 119910 isin 119909 + 119861119896+119895



119910 + 119861119896sub 119909 + 119861

119896+119895+ 119861

119896sub 119909 + 119861

119896+119895+120590 (35)


119887minus119902(119895+120590)

120593 (119909 + 119861119896+119895+120590

120582) le 1198621120593 (119910 + 119861

119896 120582) (36)


119896+119895+120590) we have

120593 (Ω cap (119909 + 119861119896+119895+120590

) 120582) ge 120593 (119910 + 119861119896 120582)

ge 119862minus1

1119887minus119902(119895+120590)

times 120593 (119909 + 119861119896+119895+120590

120582)

(37)

which implies that

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(38)



M120593(sdot120582)

119891 (119909) = sup119898isinZ

1

120593 (119909 + 119861119898 120582)

times int119909+119861119898

1003816100381610038161003816119891 (119910)1003816100381610038161003816 120593 (119910 120582) 119889119910

(39)

Thus

119909 isin R119899

119865lowast119870

119895(119909) gt 119905

sub 119909 isin R119899

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(40)




intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 sim int

R119899int

119865lowast119870

119895(119909)

0

120593 (119909 119905)119889119905

119905119889119909

sim int

infin

0

int119909isinR119899119865lowast119870

119895(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

(41)


intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲ 1198871199022119895

int

infin

0

int119909isinR119899119865lowast119870

0(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

sim 1198871199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

(42)




119895) isin


120593(119909

infin

sum

119895=1

119905119895) le 119862

infin

sum

119895=1

120593 (119909 119905119895) (43)



2


10038171003817100381710038171003817119879119873(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

le 11986210038171003817100381710038171003817M(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

(44)



119865 (119910 119896) =

1003816100381610038161003816119891 lowast 120595119896 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)(45)

with 119898119870119871


([119902(120593)]2



le 119865lowast119870

0(119909) (46)



119861119896+119895


+ then


le 119887minus119895119873

119865lowast119870

119895(119909)

(47)


where 119865lowast119870119895



119879119873(119870119871)

120595119891 (119909) le

infin

sum

119895=0

119887minus119895119873

119865lowast119870

119895(119909) (48)




intR119899120593 (119909 119879

119873(119870119871)

120595119891 (119909)) 119889119909

le

infin

sum

119895=0

119887minus119895119873119901

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲

infin

sum

119895=0

119887minus119895(119873119901minus119902

2)

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

≲ intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(49)





+and 119909 isin R119899

119891lowast0(119870119871)

119898(119909) le 119862119879

119873(119870119871)

120595119891 (119909) (50)




M(119870119871)

120595119891 (119909) le 119862[max 1 120588 (119909)]minus119872 (51)





1

loc(R119899

) capS1015840(R119899)

119891 (119909) le 119891lowast

119898(119909) le 119862119891

lowast0

119898(119909) le 119862M

119860119891 (119909) (52)

where 119891lowast0119898(119909) = sup

120601isinS119898(R119899)sup




M119860119891 (119909) = sup

119909isin119861119861isinB

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (53)




120593


120593le 119901

+



on 119871120593(R119899)


([119902(120593)]2


+

119891lowast0(1198700)

119898(119909) ≲ 119879

119873(1198700)

120595119891 (119909) (54)


+10038171003817100381710038171003817119891lowast0(1198700)

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

(55)


10038171003817100381710038171003817119891lowast0

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

(56)


SupposeM0

120595119891 isin 119871

120593


120595119891 isin

119871120593


such that

intR119899120593 (119909 119891

lowast0(119870119871)

119898(119909)) 119889119909

le 1198621intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(57)


+ For


Ω119870= 119909 isin R

119899

1198910lowast(119870119871)

119898(119909) le 119862

2M(119870119871)

120595119891 (119909) (58)

where 1198622= [2119862


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) le 2int

Ω119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909 (59)


12 we have

intΩ∁

119870

120593 (119909M(119870119871)

120595(119909)) lt 119862

minus119901

2intΩ∁

119870

120593 (119909 1198910lowast(119870119871)

119898(119909)) 119889119909

le 119862minus119901

21198621intR119899120593 (119909M

(119870119871)

120595(119909)) 119889119909

(60)





+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)


1119902


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)




Since M(119870119871)



that M120595119891 isin 119871

120593


120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)









119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)


Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)



such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861


(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)


120590 0 le 120579 le 1 and 120579 equiv 1


119895(119909) = 120579(119860


119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By



120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861


119894120577119894= 120594


119894119894



119904(R119899) denote


119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot



119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577




119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)


119894)120577119894



119892 = 119891 minus sum119894119887119894in S1015840(R119899)









120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 120572

119899) isin Z119899

+ |120572| = 120572

1+


120572

= (1205971205971199091)1205721 sdot sdot sdot (120597120597119909

119899)120572119899 Throughout the











120601 (119904119905) le 119862119904119901

120601 (119905) (3)


120593 (119909 119904119905) le 119862119904119901

120593 (119909 119905) (4)


(0infin) Let





(5)





+be two positive

numbers such that


+

(6)


+= max|120582| 120582 isin 120590(119860)



119896= 119860

119896


119896sub 119903119861

119896sub

119861119896+1

and |119861119896| = 119887


119861119896+ 119861

119895sub 119861

119895+120590 (7)

119861119896+ (119861

119896+120590)∁

sub (119861119896)∁

(8)



119860 R119899 rarr [0infin)









120588 (119909) = sum

119896isinZ

119887119896

120594119861119896+1119861119896

(119909) if 119909 = 0 or else 120588 (0) = 0 (9)


120588 (119909 + 119910) le 119887120590

(max 120588 (119909) 120588 (119910)) le 119887120590 [120588 (119909) + 120588 (119910)] (10)




119901(119860) denoted by 120593 isin A



(1infin)


sup119896isinZ

119887minus119896

int119909+119861119896

120593 (119910 119905) 119889119910


int119909+119861119896

[120593(119910 119905)]minus1(119901minus1)

119889119910

119901minus1

le 119862

(11)


and when 119901 = 1


sup119896isinZ

119887minus119896

int119909+119861119896

120593 (119910 119905) 119889119910ess sup119910isin119909+119861119896

[120593(119910 119905)]minus1

le 119862

(12)


(120593)Define A

infin(119860) = ⋃

1le119901ltinfinA119901(119860) and





119902(120593)(119860)


A119902(119860)) A

1(119860) such










119898(120593) = lfloor(119902 (120593)

119894 (120593)minus 1)

ln 119887ln 120582

minus

rfloor (14)

Clearly

120593 (119909 119905) = 119908 (119909)Φ (119905) forall119909 isin R119899

119905 isin [0infin) (15)






intR119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 lt infin (16)



R119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 le 1

(17)


S119898(R119899

) = 120601 isin S (R119899

)

sup119909isinR119899|120572|le119898+1

[1 + 120588(119909)]119898+2 1003816100381610038161003816120597

120572

120601 (119909)1003816100381610038161003816 le 1

(18)


119896

120593(119860119896



119891lowast

119898(119909) = sup

120601isinS119898(R119899)

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816119891 lowast 120601119896 (119910)1003816100381610038161003816 (19)





119898isin 119871

120593


119898119860(R119899) = 119891

lowast

119898119871120593(R119899)

When119898 = 119898(120593)119867120593


119860(R119899)










(R119899)


(R119899) sub


Proof Let 119891 isin 119867120593


0


119909(119910) = 120601(119909 minus 119910) for all

119910 isin R119899



sup119909isin1198610 119910isinR

119899

1 + 120588 (119910)

1 + 120588 (119909 minus 119910)le 119887

2120590

(20)


1003816100381610038161003816⟨119891 120601⟩1003816100381610038161003816 =

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

1003816100381610038161003816100381610038161003816100381610038161003816

119891 lowast (120595119909

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

) (119909)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)inf119909isin1198610

119891lowast

119898(119909)

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)

100381710038171003817100381710038171205941198610

10038171003817100381710038171003817

minus1

119871120593(R119899)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(21)




119898119860(R119899) is complete


Maximal Functions






M0

120595119891 (119909) = sup

119896isinZ

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

M120595119891 (119909) = sup

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119879119873

120595119891 (119909) = sup

119896isinZ


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816


+

(22)




119891 isin 119867120593

119860(R119899

) (23)

119879119873

120595119891 isin 119871

120593

(R119899

) 119873 gt[119902 (120593)]

2

119894 (120593) (24)

M120595119891 isin 119871

120593

(R119899

) (25)

M0

120595119891 isin 119871

120593

(R119899

) (26)



120593

119860(R119899) such that

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

=1003817100381710038171003817119891lowast

119898

1003817100381710038171003817119871120593(R119899) le 119862110038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198622

10038171003817100381710038171003817M0

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198623

10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198624

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(27)



120593 (119864 119905) = int119864

120593 (119909 119905) 119889119909 (28)



(119909 + 119861119896) and 119905 isin (0infin)

120593 (119909 + 119861119896 119905)

120593 (119864 119905)le 119862

1003816100381610038161003816119909 + 1198611198961003816100381610038161003816119902

|119864|119902

(29)


119905

0



119898119870119871

(119909 119896) = [max 1 120588 (119860minus119870119909)]119871

(1 + 119887minus119896minus119870

)119871 (30)


M0(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

119898119870119871

(119909 119896)

M(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)

119879119873(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

[1 + 120588 (119860minus119896 (119909 minus 119910))]119873

119898119870119871

(119910 119896)

119873 isin Z+

1198910lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M0(119870119871)

120595119891 (119909)

119891lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M(119870119871)

120595119891 (119909)

(31)

where S119898(R119899) is as in (18)




119865lowast119870

119895(119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119895+119896

119865 (119910 119896) (32)



lowast119870

119895(119909) gt 120582 is open for any


119895which




120593 (119909 isin R119899 119865lowast119870119895

(119909) gt 119905 120582)

le 1198621198871199022119895

120593 (119909 isin R119899 119865lowast1198700

(119909) gt 119905 120582)

(33)

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 le 119862119887

1199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909 (34)



119895(119909) gt 119905 there exist 119896 le 119870

and 119910 isin 119909 + 119861119896+119895



119910 + 119861119896sub 119909 + 119861

119896+119895+ 119861

119896sub 119909 + 119861

119896+119895+120590 (35)


119887minus119902(119895+120590)

120593 (119909 + 119861119896+119895+120590

120582) le 1198621120593 (119910 + 119861

119896 120582) (36)


119896+119895+120590) we have

120593 (Ω cap (119909 + 119861119896+119895+120590

) 120582) ge 120593 (119910 + 119861119896 120582)

ge 119862minus1

1119887minus119902(119895+120590)

times 120593 (119909 + 119861119896+119895+120590

120582)

(37)

which implies that

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(38)



M120593(sdot120582)

119891 (119909) = sup119898isinZ

1

120593 (119909 + 119861119898 120582)

times int119909+119861119898

1003816100381610038161003816119891 (119910)1003816100381610038161003816 120593 (119910 120582) 119889119910

(39)

Thus

119909 isin R119899

119865lowast119870

119895(119909) gt 119905

sub 119909 isin R119899

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(40)




intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 sim int

R119899int

119865lowast119870

119895(119909)

0

120593 (119909 119905)119889119905

119905119889119909

sim int

infin

0

int119909isinR119899119865lowast119870

119895(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

(41)


intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲ 1198871199022119895

int

infin

0

int119909isinR119899119865lowast119870

0(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

sim 1198871199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

(42)




119895) isin


120593(119909

infin

sum

119895=1

119905119895) le 119862

infin

sum

119895=1

120593 (119909 119905119895) (43)



2


10038171003817100381710038171003817119879119873(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

le 11986210038171003817100381710038171003817M(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

(44)



119865 (119910 119896) =

1003816100381610038161003816119891 lowast 120595119896 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)(45)

with 119898119870119871


([119902(120593)]2



le 119865lowast119870

0(119909) (46)



119861119896+119895


+ then


le 119887minus119895119873

119865lowast119870

119895(119909)

(47)


where 119865lowast119870119895



119879119873(119870119871)

120595119891 (119909) le

infin

sum

119895=0

119887minus119895119873

119865lowast119870

119895(119909) (48)




intR119899120593 (119909 119879

119873(119870119871)

120595119891 (119909)) 119889119909

le

infin

sum

119895=0

119887minus119895119873119901

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲

infin

sum

119895=0

119887minus119895(119873119901minus119902

2)

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

≲ intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(49)





+and 119909 isin R119899

119891lowast0(119870119871)

119898(119909) le 119862119879

119873(119870119871)

120595119891 (119909) (50)




M(119870119871)

120595119891 (119909) le 119862[max 1 120588 (119909)]minus119872 (51)





1

loc(R119899

) capS1015840(R119899)

119891 (119909) le 119891lowast

119898(119909) le 119862119891

lowast0

119898(119909) le 119862M

119860119891 (119909) (52)

where 119891lowast0119898(119909) = sup

120601isinS119898(R119899)sup




M119860119891 (119909) = sup

119909isin119861119861isinB

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (53)




120593


120593le 119901

+



on 119871120593(R119899)


([119902(120593)]2


+

119891lowast0(1198700)

119898(119909) ≲ 119879

119873(1198700)

120595119891 (119909) (54)


+10038171003817100381710038171003817119891lowast0(1198700)

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

(55)


10038171003817100381710038171003817119891lowast0

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

(56)


SupposeM0

120595119891 isin 119871

120593


120595119891 isin

119871120593


such that

intR119899120593 (119909 119891

lowast0(119870119871)

119898(119909)) 119889119909

le 1198621intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(57)


+ For


Ω119870= 119909 isin R

119899

1198910lowast(119870119871)

119898(119909) le 119862

2M(119870119871)

120595119891 (119909) (58)

where 1198622= [2119862


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) le 2int

Ω119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909 (59)


12 we have

intΩ∁

119870

120593 (119909M(119870119871)

120595(119909)) lt 119862

minus119901

2intΩ∁

119870

120593 (119909 1198910lowast(119870119871)

119898(119909)) 119889119909

le 119862minus119901

21198621intR119899120593 (119909M

(119870119871)

120595(119909)) 119889119909

(60)





+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)


1119902


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)




Since M(119870119871)



that M120595119891 isin 119871

120593


120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)









119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)


Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)



such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861


(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)


120590 0 le 120579 le 1 and 120579 equiv 1


119895(119909) = 120579(119860


119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By



120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861


119894120577119894= 120594


119894119894



119904(R119899) denote


119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot



119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577




119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)


119894)120577119894



119892 = 119891 minus sum119894119887119894in S1015840(R119899)









120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and when 119901 = 1


sup119896isinZ

119887minus119896

int119909+119861119896

120593 (119910 119905) 119889119910ess sup119910isin119909+119861119896

[120593(119910 119905)]minus1

le 119862

(12)


(120593)Define A

infin(119860) = ⋃

1le119901ltinfinA119901(119860) and





119902(120593)(119860)


A119902(119860)) A

1(119860) such










119898(120593) = lfloor(119902 (120593)

119894 (120593)minus 1)

ln 119887ln 120582

minus

rfloor (14)

Clearly

120593 (119909 119905) = 119908 (119909)Φ (119905) forall119909 isin R119899

119905 isin [0infin) (15)






intR119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 lt infin (16)



R119899120593(119909

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582) 119889119909 le 1

(17)


S119898(R119899

) = 120601 isin S (R119899

)

sup119909isinR119899|120572|le119898+1

[1 + 120588(119909)]119898+2 1003816100381610038161003816120597

120572

120601 (119909)1003816100381610038161003816 le 1

(18)


119896

120593(119860119896



119891lowast

119898(119909) = sup

120601isinS119898(R119899)

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816119891 lowast 120601119896 (119910)1003816100381610038161003816 (19)





119898isin 119871

120593


119898119860(R119899) = 119891

lowast

119898119871120593(R119899)

When119898 = 119898(120593)119867120593


119860(R119899)










(R119899)


(R119899) sub


Proof Let 119891 isin 119867120593


0


119909(119910) = 120601(119909 minus 119910) for all

119910 isin R119899



sup119909isin1198610 119910isinR

119899

1 + 120588 (119910)

1 + 120588 (119909 minus 119910)le 119887

2120590

(20)


1003816100381610038161003816⟨119891 120601⟩1003816100381610038161003816 =

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

1003816100381610038161003816100381610038161003816100381610038161003816

119891 lowast (120595119909

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

) (119909)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)inf119909isin1198610

119891lowast

119898(119909)

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)

100381710038171003817100381710038171205941198610

10038171003817100381710038171003817

minus1

119871120593(R119899)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(21)




119898119860(R119899) is complete


Maximal Functions






M0

120595119891 (119909) = sup

119896isinZ

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

M120595119891 (119909) = sup

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119879119873

120595119891 (119909) = sup

119896isinZ


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816


+

(22)




119891 isin 119867120593

119860(R119899

) (23)

119879119873

120595119891 isin 119871

120593

(R119899

) 119873 gt[119902 (120593)]

2

119894 (120593) (24)

M120595119891 isin 119871

120593

(R119899

) (25)

M0

120595119891 isin 119871

120593

(R119899

) (26)



120593

119860(R119899) such that

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

=1003817100381710038171003817119891lowast

119898

1003817100381710038171003817119871120593(R119899) le 119862110038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198622

10038171003817100381710038171003817M0

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198623

10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198624

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(27)



120593 (119864 119905) = int119864

120593 (119909 119905) 119889119909 (28)



(119909 + 119861119896) and 119905 isin (0infin)

120593 (119909 + 119861119896 119905)

120593 (119864 119905)le 119862

1003816100381610038161003816119909 + 1198611198961003816100381610038161003816119902

|119864|119902

(29)


119905

0



119898119870119871

(119909 119896) = [max 1 120588 (119860minus119870119909)]119871

(1 + 119887minus119896minus119870

)119871 (30)


M0(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

119898119870119871

(119909 119896)

M(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)

119879119873(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

[1 + 120588 (119860minus119896 (119909 minus 119910))]119873

119898119870119871

(119910 119896)

119873 isin Z+

1198910lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M0(119870119871)

120595119891 (119909)

119891lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M(119870119871)

120595119891 (119909)

(31)

where S119898(R119899) is as in (18)




119865lowast119870

119895(119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119895+119896

119865 (119910 119896) (32)



lowast119870

119895(119909) gt 120582 is open for any


119895which




120593 (119909 isin R119899 119865lowast119870119895

(119909) gt 119905 120582)

le 1198621198871199022119895

120593 (119909 isin R119899 119865lowast1198700

(119909) gt 119905 120582)

(33)

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 le 119862119887

1199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909 (34)



119895(119909) gt 119905 there exist 119896 le 119870

and 119910 isin 119909 + 119861119896+119895



119910 + 119861119896sub 119909 + 119861

119896+119895+ 119861

119896sub 119909 + 119861

119896+119895+120590 (35)


119887minus119902(119895+120590)

120593 (119909 + 119861119896+119895+120590

120582) le 1198621120593 (119910 + 119861

119896 120582) (36)


119896+119895+120590) we have

120593 (Ω cap (119909 + 119861119896+119895+120590

) 120582) ge 120593 (119910 + 119861119896 120582)

ge 119862minus1

1119887minus119902(119895+120590)

times 120593 (119909 + 119861119896+119895+120590

120582)

(37)

which implies that

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(38)



M120593(sdot120582)

119891 (119909) = sup119898isinZ

1

120593 (119909 + 119861119898 120582)

times int119909+119861119898

1003816100381610038161003816119891 (119910)1003816100381610038161003816 120593 (119910 120582) 119889119910

(39)

Thus

119909 isin R119899

119865lowast119870

119895(119909) gt 119905

sub 119909 isin R119899

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(40)




intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 sim int

R119899int

119865lowast119870

119895(119909)

0

120593 (119909 119905)119889119905

119905119889119909

sim int

infin

0

int119909isinR119899119865lowast119870

119895(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

(41)


intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲ 1198871199022119895

int

infin

0

int119909isinR119899119865lowast119870

0(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

sim 1198871199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

(42)




119895) isin


120593(119909

infin

sum

119895=1

119905119895) le 119862

infin

sum

119895=1

120593 (119909 119905119895) (43)



2


10038171003817100381710038171003817119879119873(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

le 11986210038171003817100381710038171003817M(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

(44)



119865 (119910 119896) =

1003816100381610038161003816119891 lowast 120595119896 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)(45)

with 119898119870119871


([119902(120593)]2



le 119865lowast119870

0(119909) (46)



119861119896+119895


+ then


le 119887minus119895119873

119865lowast119870

119895(119909)

(47)


where 119865lowast119870119895



119879119873(119870119871)

120595119891 (119909) le

infin

sum

119895=0

119887minus119895119873

119865lowast119870

119895(119909) (48)




intR119899120593 (119909 119879

119873(119870119871)

120595119891 (119909)) 119889119909

le

infin

sum

119895=0

119887minus119895119873119901

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲

infin

sum

119895=0

119887minus119895(119873119901minus119902

2)

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

≲ intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(49)





+and 119909 isin R119899

119891lowast0(119870119871)

119898(119909) le 119862119879

119873(119870119871)

120595119891 (119909) (50)




M(119870119871)

120595119891 (119909) le 119862[max 1 120588 (119909)]minus119872 (51)





1

loc(R119899

) capS1015840(R119899)

119891 (119909) le 119891lowast

119898(119909) le 119862119891

lowast0

119898(119909) le 119862M

119860119891 (119909) (52)

where 119891lowast0119898(119909) = sup

120601isinS119898(R119899)sup




M119860119891 (119909) = sup

119909isin119861119861isinB

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (53)




120593


120593le 119901

+



on 119871120593(R119899)


([119902(120593)]2


+

119891lowast0(1198700)

119898(119909) ≲ 119879

119873(1198700)

120595119891 (119909) (54)


+10038171003817100381710038171003817119891lowast0(1198700)

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

(55)


10038171003817100381710038171003817119891lowast0

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

(56)


SupposeM0

120595119891 isin 119871

120593


120595119891 isin

119871120593


such that

intR119899120593 (119909 119891

lowast0(119870119871)

119898(119909)) 119889119909

le 1198621intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(57)


+ For


Ω119870= 119909 isin R

119899

1198910lowast(119870119871)

119898(119909) le 119862

2M(119870119871)

120595119891 (119909) (58)

where 1198622= [2119862


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) le 2int

Ω119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909 (59)


12 we have

intΩ∁

119870

120593 (119909M(119870119871)

120595(119909)) lt 119862

minus119901

2intΩ∁

119870

120593 (119909 1198910lowast(119870119871)

119898(119909)) 119889119909

le 119862minus119901

21198621intR119899120593 (119909M

(119870119871)

120595(119909)) 119889119909

(60)





+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)


1119902


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)




Since M(119870119871)



that M120595119891 isin 119871

120593


120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)









119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)


Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)



such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861


(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)


120590 0 le 120579 le 1 and 120579 equiv 1


119895(119909) = 120579(119860


119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By



120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861


119894120577119894= 120594


119894119894



119904(R119899) denote


119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot



119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577




119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)


119894)120577119894



119892 = 119891 minus sum119894119887119894in S1015840(R119899)









120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sup119909isin1198610 119910isinR

119899

1 + 120588 (119910)

1 + 120588 (119909 minus 119910)le 119887

2120590

(20)


1003816100381610038161003816⟨119891 120601⟩1003816100381610038161003816 =

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

1003816100381610038161003816100381610038161003816100381610038161003816

119891 lowast (120595119909

10038171003817100381710038171205951199091003817100381710038171003817S119898(R

119899)

) (119909)

1003816100381610038161003816100381610038161003816100381610038161003816

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)inf119909isin1198610

119891lowast

119898(119909)

le 1198872120590(119898+2)1003817100381710038171003817120601

1003817100381710038171003817S119898(R119899)

100381710038171003817100381710038171205941198610

10038171003817100381710038171003817

minus1

119871120593(R119899)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(21)




119898119860(R119899) is complete


Maximal Functions






M0

120595119891 (119909) = sup

119896isinZ

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

M120595119891 (119909) = sup

119896isinZ

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119879119873

120595119891 (119909) = sup

119896isinZ


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816


+

(22)




119891 isin 119867120593

119860(R119899

) (23)

119879119873

120595119891 isin 119871

120593

(R119899

) 119873 gt[119902 (120593)]

2

119894 (120593) (24)

M120595119891 isin 119871

120593

(R119899

) (25)

M0

120595119891 isin 119871

120593

(R119899

) (26)



120593

119860(R119899) such that

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

=1003817100381710038171003817119891lowast

119898

1003817100381710038171003817119871120593(R119899) le 119862110038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198622

10038171003817100381710038171003817M0

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198623

10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

le 1198624

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(27)



120593 (119864 119905) = int119864

120593 (119909 119905) 119889119909 (28)



(119909 + 119861119896) and 119905 isin (0infin)

120593 (119909 + 119861119896 119905)

120593 (119864 119905)le 119862

1003816100381610038161003816119909 + 1198611198961003816100381610038161003816119902

|119864|119902

(29)


119905

0



119898119870119871

(119909 119896) = [max 1 120588 (119860minus119870119909)]119871

(1 + 119887minus119896minus119870

)119871 (30)


M0(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

1003816100381610038161003816120595119896 lowast 119891 (119909)1003816100381610038161003816

119898119870119871

(119909 119896)

M(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119896

1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)

119879119873(119870119871)

120595119891 (119909) = sup

119896isinZ119896le119870


1003816100381610038161003816120595119896 lowast 119891 (119910)1003816100381610038161003816

[1 + 120588 (119860minus119896 (119909 minus 119910))]119873

119898119870119871

(119910 119896)

119873 isin Z+

1198910lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M0(119870119871)

120595119891 (119909)

119891lowast(119870119871)

119898(119909) = sup

120595isinS119898(R119899)

M(119870119871)

120595119891 (119909)

(31)

where S119898(R119899) is as in (18)




119865lowast119870

119895(119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119895+119896

119865 (119910 119896) (32)



lowast119870

119895(119909) gt 120582 is open for any


119895which




120593 (119909 isin R119899 119865lowast119870119895

(119909) gt 119905 120582)

le 1198621198871199022119895

120593 (119909 isin R119899 119865lowast1198700

(119909) gt 119905 120582)

(33)

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 le 119862119887

1199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909 (34)



119895(119909) gt 119905 there exist 119896 le 119870

and 119910 isin 119909 + 119861119896+119895



119910 + 119861119896sub 119909 + 119861

119896+119895+ 119861

119896sub 119909 + 119861

119896+119895+120590 (35)


119887minus119902(119895+120590)

120593 (119909 + 119861119896+119895+120590

120582) le 1198621120593 (119910 + 119861

119896 120582) (36)


119896+119895+120590) we have

120593 (Ω cap (119909 + 119861119896+119895+120590

) 120582) ge 120593 (119910 + 119861119896 120582)

ge 119862minus1

1119887minus119902(119895+120590)

times 120593 (119909 + 119861119896+119895+120590

120582)

(37)

which implies that

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(38)



M120593(sdot120582)

119891 (119909) = sup119898isinZ

1

120593 (119909 + 119861119898 120582)

times int119909+119861119898

1003816100381610038161003816119891 (119910)1003816100381610038161003816 120593 (119910 120582) 119889119910

(39)

Thus

119909 isin R119899

119865lowast119870

119895(119909) gt 119905

sub 119909 isin R119899

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(40)




intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 sim int

R119899int

119865lowast119870

119895(119909)

0

120593 (119909 119905)119889119905

119905119889119909

sim int

infin

0

int119909isinR119899119865lowast119870

119895(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

(41)


intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲ 1198871199022119895

int

infin

0

int119909isinR119899119865lowast119870

0(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

sim 1198871199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

(42)




119895) isin


120593(119909

infin

sum

119895=1

119905119895) le 119862

infin

sum

119895=1

120593 (119909 119905119895) (43)



2


10038171003817100381710038171003817119879119873(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

le 11986210038171003817100381710038171003817M(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

(44)



119865 (119910 119896) =

1003816100381610038161003816119891 lowast 120595119896 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)(45)

with 119898119870119871


([119902(120593)]2



le 119865lowast119870

0(119909) (46)



119861119896+119895


+ then


le 119887minus119895119873

119865lowast119870

119895(119909)

(47)


where 119865lowast119870119895



119879119873(119870119871)

120595119891 (119909) le

infin

sum

119895=0

119887minus119895119873

119865lowast119870

119895(119909) (48)




intR119899120593 (119909 119879

119873(119870119871)

120595119891 (119909)) 119889119909

le

infin

sum

119895=0

119887minus119895119873119901

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲

infin

sum

119895=0

119887minus119895(119873119901minus119902

2)

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

≲ intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(49)





+and 119909 isin R119899

119891lowast0(119870119871)

119898(119909) le 119862119879

119873(119870119871)

120595119891 (119909) (50)




M(119870119871)

120595119891 (119909) le 119862[max 1 120588 (119909)]minus119872 (51)





1

loc(R119899

) capS1015840(R119899)

119891 (119909) le 119891lowast

119898(119909) le 119862119891

lowast0

119898(119909) le 119862M

119860119891 (119909) (52)

where 119891lowast0119898(119909) = sup

120601isinS119898(R119899)sup




M119860119891 (119909) = sup

119909isin119861119861isinB

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (53)




120593


120593le 119901

+



on 119871120593(R119899)


([119902(120593)]2


+

119891lowast0(1198700)

119898(119909) ≲ 119879

119873(1198700)

120595119891 (119909) (54)


+10038171003817100381710038171003817119891lowast0(1198700)

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

(55)


10038171003817100381710038171003817119891lowast0

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

(56)


SupposeM0

120595119891 isin 119871

120593


120595119891 isin

119871120593


such that

intR119899120593 (119909 119891

lowast0(119870119871)

119898(119909)) 119889119909

le 1198621intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(57)


+ For


Ω119870= 119909 isin R

119899

1198910lowast(119870119871)

119898(119909) le 119862

2M(119870119871)

120595119891 (119909) (58)

where 1198622= [2119862


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) le 2int

Ω119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909 (59)


12 we have

intΩ∁

119870

120593 (119909M(119870119871)

120595(119909)) lt 119862

minus119901

2intΩ∁

119870

120593 (119909 1198910lowast(119870119871)

119898(119909)) 119889119909

le 119862minus119901

21198621intR119899120593 (119909M

(119870119871)

120595(119909)) 119889119909

(60)





+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)


1119902


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)




Since M(119870119871)



that M120595119891 isin 119871

120593


120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)









119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)


Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)



such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861


(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)


120590 0 le 120579 le 1 and 120579 equiv 1


119895(119909) = 120579(119860


119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By



120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861


119894120577119894= 120594


119894119894



119904(R119899) denote


119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot



119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577




119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)


119894)120577119894



119892 = 119891 minus sum119894119887119894in S1015840(R119899)









120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119865lowast119870

119895(119909) = sup

119896isinZ119896le119870

sup119910isin119909+119861119895+119896

119865 (119910 119896) (32)



lowast119870

119895(119909) gt 120582 is open for any


119895which




120593 (119909 isin R119899 119865lowast119870119895

(119909) gt 119905 120582)

le 1198621198871199022119895

120593 (119909 isin R119899 119865lowast1198700

(119909) gt 119905 120582)

(33)

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 le 119862119887

1199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909 (34)



119895(119909) gt 119905 there exist 119896 le 119870

and 119910 isin 119909 + 119861119896+119895



119910 + 119861119896sub 119909 + 119861

119896+119895+ 119861

119896sub 119909 + 119861

119896+119895+120590 (35)


119887minus119902(119895+120590)

120593 (119909 + 119861119896+119895+120590

120582) le 1198621120593 (119910 + 119861

119896 120582) (36)


119896+119895+120590) we have

120593 (Ω cap (119909 + 119861119896+119895+120590

) 120582) ge 120593 (119910 + 119861119896 120582)

ge 119862minus1

1119887minus119902(119895+120590)

times 120593 (119909 + 119861119896+119895+120590

120582)

(37)

which implies that

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(38)



M120593(sdot120582)

119891 (119909) = sup119898isinZ

1

120593 (119909 + 119861119898 120582)

times int119909+119861119898

1003816100381610038161003816119891 (119910)1003816100381610038161003816 120593 (119910 120582) 119889119910

(39)

Thus

119909 isin R119899

119865lowast119870

119895(119909) gt 119905

sub 119909 isin R119899

M120593(sdot120582)

(120594Ω) (119909) ge 119862

minus1

1119887minus119902(119895+120590)

(40)




intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909 sim int

R119899int

119865lowast119870

119895(119909)

0

120593 (119909 119905)119889119905

119905119889119909

sim int

infin

0

int119909isinR119899119865lowast119870

119895(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

(41)


intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲ 1198871199022119895

int

infin

0

int119909isinR119899119865lowast119870

0(119909)gt119905

120593 (119909 119905) 119889119909119889119905

119905

sim 1198871199022119895

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

(42)




119895) isin


120593(119909

infin

sum

119895=1

119905119895) le 119862

infin

sum

119895=1

120593 (119909 119905119895) (43)



2


10038171003817100381710038171003817119879119873(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

le 11986210038171003817100381710038171003817M(119870119871)

12059511989110038171003817100381710038171003817119871120593(R119899)

(44)



119865 (119910 119896) =

1003816100381610038161003816119891 lowast 120595119896 (119910)1003816100381610038161003816

119898119870119871

(119910 119896)(45)

with 119898119870119871


([119902(120593)]2



le 119865lowast119870

0(119909) (46)



119861119896+119895


+ then


le 119887minus119895119873

119865lowast119870

119895(119909)

(47)


where 119865lowast119870119895



119879119873(119870119871)

120595119891 (119909) le

infin

sum

119895=0

119887minus119895119873

119865lowast119870

119895(119909) (48)




intR119899120593 (119909 119879

119873(119870119871)

120595119891 (119909)) 119889119909

le

infin

sum

119895=0

119887minus119895119873119901

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲

infin

sum

119895=0

119887minus119895(119873119901minus119902

2)

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

≲ intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(49)





+and 119909 isin R119899

119891lowast0(119870119871)

119898(119909) le 119862119879

119873(119870119871)

120595119891 (119909) (50)




M(119870119871)

120595119891 (119909) le 119862[max 1 120588 (119909)]minus119872 (51)





1

loc(R119899

) capS1015840(R119899)

119891 (119909) le 119891lowast

119898(119909) le 119862119891

lowast0

119898(119909) le 119862M

119860119891 (119909) (52)

where 119891lowast0119898(119909) = sup

120601isinS119898(R119899)sup




M119860119891 (119909) = sup

119909isin119861119861isinB

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (53)




120593


120593le 119901

+



on 119871120593(R119899)


([119902(120593)]2


+

119891lowast0(1198700)

119898(119909) ≲ 119879

119873(1198700)

120595119891 (119909) (54)


+10038171003817100381710038171003817119891lowast0(1198700)

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

(55)


10038171003817100381710038171003817119891lowast0

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

(56)


SupposeM0

120595119891 isin 119871

120593


120595119891 isin

119871120593


such that

intR119899120593 (119909 119891

lowast0(119870119871)

119898(119909)) 119889119909

le 1198621intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(57)


+ For


Ω119870= 119909 isin R

119899

1198910lowast(119870119871)

119898(119909) le 119862

2M(119870119871)

120595119891 (119909) (58)

where 1198622= [2119862


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) le 2int

Ω119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909 (59)


12 we have

intΩ∁

119870

120593 (119909M(119870119871)

120595(119909)) lt 119862

minus119901

2intΩ∁

119870

120593 (119909 1198910lowast(119870119871)

119898(119909)) 119889119909

le 119862minus119901

21198621intR119899120593 (119909M

(119870119871)

120595(119909)) 119889119909

(60)





+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)


1119902


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)




Since M(119870119871)



that M120595119891 isin 119871

120593


120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)









119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)


Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)



such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861


(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)


120590 0 le 120579 le 1 and 120579 equiv 1


119895(119909) = 120579(119860


119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By



120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861


119894120577119894= 120594


119894119894



119904(R119899) denote


119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot



119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577




119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)


119894)120577119894



119892 = 119891 minus sum119894119887119894in S1015840(R119899)









120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119865lowast119870119895



119879119873(119870119871)

120595119891 (119909) le

infin

sum

119895=0

119887minus119895119873

119865lowast119870

119895(119909) (48)




intR119899120593 (119909 119879

119873(119870119871)

120595119891 (119909)) 119889119909

le

infin

sum

119895=0

119887minus119895119873119901

intR119899120593 (119909 119865

lowast119870

119895(119909)) 119889119909

≲

infin

sum

119895=0

119887minus119895(119873119901minus119902

2)

intR119899120593 (119909 119865

lowast119870

0(119909)) 119889119909

≲ intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(49)





+and 119909 isin R119899

119891lowast0(119870119871)

119898(119909) le 119862119879

119873(119870119871)

120595119891 (119909) (50)




M(119870119871)

120595119891 (119909) le 119862[max 1 120588 (119909)]minus119872 (51)





1

loc(R119899

) capS1015840(R119899)

119891 (119909) le 119891lowast

119898(119909) le 119862119891

lowast0

119898(119909) le 119862M

119860119891 (119909) (52)

where 119891lowast0119898(119909) = sup

120601isinS119898(R119899)sup




M119860119891 (119909) = sup

119909isin119861119861isinB

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (53)




120593


120593le 119901

+



on 119871120593(R119899)


([119902(120593)]2


+

119891lowast0(1198700)

119898(119909) ≲ 119879

119873(1198700)

120595119891 (119909) (54)


+10038171003817100381710038171003817119891lowast0(1198700)

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M(1198700)

12059511989110038171003817100381710038171003817119871120593(R119899)

(55)


10038171003817100381710038171003817119891lowast0

119898

10038171003817100381710038171003817119871120593(R119899)≲10038171003817100381710038171003817119879119873

12059511989110038171003817100381710038171003817119871120593(R119899)

≲10038171003817100381710038171003817M12059511989110038171003817100381710038171003817119871120593(R119899)

(56)


SupposeM0

120595119891 isin 119871

120593


120595119891 isin

119871120593


such that

intR119899120593 (119909 119891

lowast0(119870119871)

119898(119909)) 119889119909

le 1198621intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

(57)


+ For


Ω119870= 119909 isin R

119899

1198910lowast(119870119871)

119898(119909) le 119862

2M(119870119871)

120595119891 (119909) (58)

where 1198622= [2119862


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) le 2int

Ω119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909 (59)


12 we have

intΩ∁

119870

120593 (119909M(119870119871)

120595(119909)) lt 119862

minus119901

2intΩ∁

119870

120593 (119909 1198910lowast(119870119871)

119898(119909)) 119889119909

le 119862minus119901

21198621intR119899120593 (119909M

(119870119871)

120595(119909)) 119889119909

(60)





+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)


1119902


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)




Since M(119870119871)



that M120595119891 isin 119871

120593


120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)









119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)


Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)



such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861


(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)


120590 0 le 120579 le 1 and 120579 equiv 1


119895(119909) = 120579(119860


119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By



120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861


119894120577119894= 120594


119894119894



119904(R119899) denote


119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot



119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577




119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)


119894)120577119894



119892 = 119891 minus sum119894119887119894in S1015840(R119899)









120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













+and

119909 isin Ω119870

M(119870119871)

120595119891 (119909) le 119862

3[M

119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

(61)


1119902


intR119899120593 (119909M

(119870119871)

120595119891 (119909)) 119889119909

le 2intΩ119870

120593 (119909M(119870119871)

120595119891 (119909)) 119889119909

le 21198623intΩ119870

120593(119909 [M119860([M

0(119870119871)

120595119891]119902

) (119909)]1119902

)119889119909

le 1198624intR119899120593 (119909M

0(119870119871)

120595119891 (119909)) 119889119909

(62)




Since M(119870119871)



that M120595119891 isin 119871

120593


120595119891119871120593(R119899)

le 1198624M0

120595119891119871120593(R119899)









119898(119909)gt120582

120593 (119909 119905) 119889119909 lt infin (63)


Ω = 119909 isin R119899

119891lowast

119898(119909) gt 120582 (64)



such that

Ω = cup119895(119909119895+ 119861

ℓ119895) (65)

(119909119894+ 119861

ℓ119894minus2120590) cap (119909

119895+ 119861


(119909119895+ 119861

ℓ119895+4120590) cap Ω

∁

= 0 (119909119895+ 119861

ℓ119895+4120590+1) cap Ω

∁

= 0 forall119895

(67)

(119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 implies that

10038161003816100381610038161003816ℓ119894minus ℓ119895

10038161003816100381610038161003816le 120590

(68)

119895 (119909119894+ 119861

ℓ119894+2120590) cap (119909

119895+ 119861

ℓ119895+2120590) = 0 le 119871 forall119894 (69)


120590 0 le 120579 le 1 and 120579 equiv 1


119895(119909) = 120579(119860


119909119895)) Clearly supp 120579

119895sub 119909

119895+ 119861

ℓ119895+120590and 120579

119895equiv 1 on 119909

119895+ 119861

ℓ119895 By



120577119894(119909) =

120579119894(119909)

sum119895120579119895(119909)

(70)

Then 120577119894isin S(R119899) supp 120577

119894sub 119909

119894+ 119861

ℓ119894+120590 0 le 120577

119894le 1 120577

119894equiv 1 on

119909119894+ 119861


119894120577119894= 120594


119894119894



119904(R119899) denote


119904(R119899) let

119875119894= [

1

intR119899120577119894(119909) 119889119909

intR119899|119875 (119909)|

2

120577119894(119909) 119889119909]

12

(71)

Then (P119904(R119899) sdot



119904(R119899) via 119876 997891rarr (1 int

R119899120577119894(119909)119889119909)⟨119891 119876120577




119904(R119899)

1

intR119899120577119894(119909) 119889119909

⟨119891119876120577119894⟩ =

1

intR119899120577119894(119909) 119889119909

⟨119875119894 119876120577

119894⟩

=1

intR119899120577119894(119909) 119889119909

intR119899119875119894(119909)119876 (119909) 120577

119894(119909) 119889119909

(72)


119894)120577119894



119892 = 119891 minus sum119894119887119894in S1015840(R119899)









120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120593(sdot1)(R119899) cap 119867

120593

119860(R119899) in

119867120593





sup|120572|le119898


10038161003816100381610038161003816120597120572

[120577119894(119860ℓ

sdot)] (119909)10038161003816100381610038161003816le 119862

1 (73)




1003816100381610038161003816119875119894 (119910) 120577119894 (119910)1003816100381610038161003816 le 1198622 sup

119910isin(119909119894+119861ℓ119894+4120590+1)capΩ∁

119891lowast

119898(119910) le 119862

2120582 (74)



ℓ119894+2120590 (119887119894)lowast

119898(119909) le

1198623119891lowast

119898(119909)


+ 119894 and

119909 isin 119909119894+ 119861

119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590

(119887119894)lowast

119898(119909) le 119862

4120582(120582

minus)minus119905(119904+1)



120593

119898119860(R119899)

120582 isin (0infin) and 119894

intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 le 119862

5int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909 (75)


119898119860(R119899) and

intR119899120593(119909(sum

119894

119887119894)

lowast

119898

(119909))119889119909 le 1198711198625intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(76)



intR119899120593 (119909 (119887

119894)lowast

119898(119909)) 119889119909 ≲int

119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

+ int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

(77)


119887minus(119902(120593)+120578)

(120582minus)(119904+1)119901


119902(120593)+120578(119860)


119894+ 119861

ℓ119894+2120590

we have

int(119909119894+119861ℓ119894+2120590

)∁

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

=

infin

sum

119905=0

int119909119894+(119861119905+ℓ119894+2120590+1

119861119905+ℓ119894+2120590)

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909

≲ 120593 (119909119894+ 119861

ℓ119894+2120590 120582)

infin

sum

119905=0

119887minus[119902(120593)+120578]

(120582minus)(119904+1)119901

minus119905

≲ int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

(78)


intR119899sum

119894

120593 (119909 (119887119894)lowast

119898(119909)) 119889119909 ≲ sum

119894

int119909119894+119861ℓ119894+2120590

120593 (119909 119891lowast

119898(119909)) 119889119909

≲ intΩ

120593 (119909 119891lowast

119898(119909)) 119889119909

(79)



119894119887119894converges in 119867120593

119898119860(R119899) So



119898le sum

119894(119887119894)lowast

119898 From this






120593(sdot1)(R119899) then


119902



sum119894|119887119894|119871119902

120593(sdot1)(R119899) le 1198626119891119871

119902

120593(sdot1)(R119899)





119892lowast

119898(119909) le 119862

7120582sum

119894

(120582minus)minus119905119894(119909)(119904+1)

+ 119891lowast

119898(119909) 120594

Ω∁ (119909) (80)

where

119905119894(119909) =

120581119894 if 119909 isin 119909

119894+ (119861

120581119894+ℓ119894+2120590+1 119861120581119894+ℓ119894+2120590

)

for some 120581119894ge 0

0 otherwise(81)




120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120593

119898119860(R119899)

then 119892lowast

119898isin 119871

119902




119898(119909)]

119902

120593 (119909 1) 119889119909

le 1198628120582119902

(max 11205821

120582119901)int

R119899120593 (119909 119891

lowast

119898(119909)) 119889119909

(82)


120593(sdot1)(R119899) then 119892 isin 119871

infin


9 independent of


Proof Since 119891 isin 119867120593


sum119894119887119894converges in 119867

120593




119898(119909)]

119902

120593 (119909 1) 119889119909

≲ 120582119902

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119904+1)]

119902

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(83)



minus)119898+1

gt 119887119902(120593) More-


119905+ℓ119894+2120590+1 119861119905+ℓ119894+2120590


we find that

119887minus119905

≲1

10038161003816100381610038161003816119909119894+ 119861

119905+ℓ119894+2120590+1

10038161003816100381610038161003816

int119909119894+119861119905+ℓ119894+2120590+1

120594119909119894+119861ℓ119894

(119910) 119889119910

≲ M119860(120594119909119894+119861ℓ119894

) (119909)

(84)


(ℓ119902(120593)


119860(see [42Theorem 25]) (65) and (69)

it follows that

intR119899[sum

119894

(120582minus)minus119905119894(119909)(119898+1)]

119902

120593 (119909 1) 119889119909

le intR119899[sum

119894

119887minus119905119894(119909)119902(120593)]

119902

120593 (119909 1) 119889119909

≲ intR119899

(sum

119894

[M119860(120594119909119894+119861ℓ119894

) (119909)]119902(120593)

)

1119902(120593)

119902119902(120593)

times 120593 (119909 1) 119889119909

≲ intR119899[sum

119894

(120594119909119894+119861ℓ119894

)119902(120593)

]

119902

120593 (119909 1) 119889119909

≲ intΩ

120593 (119909 1) 119889119909

(85)

and hence


119898(119909)]

119902

120593 (119909 1) 119889119909 ≲ 120582119902

intΩ

120593 (119909 1) 119889119909

+ intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

(86)


we find that

intΩ

120593 (119909 1) 119889119909 ≲ (max 11205821

120582119901)int

Ω

120593 (119909 119891lowast

119898(119909)) 119889119909 (87)


120593 (119909 120582) ≲ 120593 (119909 119891lowast

119898(119909))

120582119902

[119891lowast119898(119909)]

119902 (88)

we see that

intΩ∁

[119891lowast

119898(119909)]

119902

120593 (119909 1) 119889119909

≲ (max 11205821

120582119901)int

Ω∁

[119891lowast

119898(119909)]

119902

120593 (119909 120582) 119889119909

≲ 120582119902

(max 11205821

120582119901)int

Ω∁

120593 (119909 119891lowast

119898(119909)) 119889119909

(89)



120593(sdot1)(R119899) then 119892 and 119887

119894119894


120593(sdot1)(R119899) and



119892 = 119891 minussum

119894

119887119894= 119891(1 minussum

119894

120577119894) +sum

119894

119875119894120577119894

= 119891120594Ω∁ +sum

119894

119875119894120577119894

(90)


lowast






119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) is

dense in119867120593119898119860

(R119899)

Proof Let 119891 isin 119867120593


119892120582

+ sum119894119887120582







sum

119894

119887120582

119894

1003817100381710038171003817100381710038171003817100381710038171003817119867120593

119898119860(R119899)

≲ int

119909isinR119899119891lowast119898(119909)gt120582

120593 (119909 119891lowast

119898(119909)) 119889119909 997888rarr 0

(91)




119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119898)120582

isin 119871119902












10038171003817100381710038171198911003817100381710038171003817119871119902

120593(119861)=


[1

120593 (119861 119905)intR119899

1003816100381610038161003816119891(119909)1003816100381610038161003816119902

120593 (119909 119905) 119889119909]

1119902

ltinfin



(92)


119871119902

120593(119861)) is a Banach




+such that 119904 ge 119898(120593) with

119898(120593) as in (14)



(b) 119886119871119902

120593(119861)le 120594

119861minus1

119871120593(R119899)

(c) intR119899119886(119909)119909

120572

119889119909 = 0 for any |120572| le 119904




119895119886119895inS1015840(R119899) where 119886




sum

119895

120593(119909119895+ 119861

ℓ11989510038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

) lt infin (93)

Define

Λ119902(119886

119895)

= inf


119895

120593(119909119895+ 119861

ℓ119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

120582) le 1

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860(R119899)

= inf

Λ119902(119886

119895) 119891 = sum

119895

119886119895in S

1015840

(R119899

)

(94)



119895120582119895119886119895in S1015840(R119899) where 119886

119895119895are


ℓ119895119895 and

10038171003817100381710038171198911003817100381710038171003817120593119902119904

119860(R119899)

= inf

Λ119902(120582

119895) 119891 = sum

119895

120582119895119886119895in S

1015840

(R119899

)

lt infin

(95)


Λ119902(120582

119895119895

)

= inf


sum

119895

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

(96)


(R119899) and the space 119867120593119902119904119860

(R119899)



119886119895119895 and 120582


119902(120582


119895=


119902(119886119895) ≲ Λ

119902(120582

119895) lt infin


119902(119886119895) lt infin

by defining

120582119895=10038171003817100381710038171003817119886119895

10038171003817100381710038171003817119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817119871120593(R119899)

119886119895= 119886

119895

10038171003817100381710038171003817119886119895

10038171003817100381710038171003817

minus1

119871119902

120593(119909119895+119861ℓ119895)

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

(97)


119902(120582

119895) = Λ

119902(119886119895) lt infin Thus








119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119860(R119899)




0+ 119861

119895

intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 le 119862120593 (119909

0+ 119861

119895 119886

119871119902

120593(1199090+119861119895)) (98)

and hence 119886119867120593

119898119860(R119899) le 119862


intR119899120593 (119909 119886

lowast

119898(119909)) 119889119909 = int

1199090+119861119895+120590

120593 (119909 119886lowast

119898(119909)) 119889119909

+ int(1199090+119861119895+120590)

∁



119895 119886

119871119902

120593(1199090+119861119895)) is


+and 119909 isin 119909

0+

(119861119895+120590+ℓ+1

119861119895+120590+ℓ

)

119886lowast

119898(119909) ≲ 119886

119871119902

120593(1199090+119861119895)[119887(120582

minus)119904+1

]minusℓ

(100)





II ≲infin

sum

ℓ=0

int1199090+(119861119895+ℓ+120590+1119861119895+ℓ+120590)

[119887(120582minus)119904+1

]minusℓ119901

times 120593 (119909 119886119871119902

120593(1199090+119861119895)) 119889119909

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

times

infin

sum

ℓ=0

[119887minus119902+119901

(120582minus)(119904+1)119901

]minusℓ

≲ 120593 (1199090+ 119861

119895 119886

119871119902

120593(1199090+119861119895))

(101)



119902(119860) and

int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119910

11199021015840

le119887119895

[120593 (1199090+ 119861

119895 120582)]

1119902

(102)

we obtain

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816 119889119910 le int

1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816119902

120593(119910 120582)119889119910

1119902

times (int1199090+119861119895

[120593 (119910 120582)]minus1199021015840119902

119889119909)

11199021015840

≲ 119887119895

119886119871119902

120593(1199090+119861119895)

(103)

Let 119909 isin 1199090+ (119861

119895+ℓ+120590+1 119861119895+ℓ+120590


119895 + 119896 gt 0 and 119910 isin 1199090+ 119861

119895 we have 120588(119860119896(119909 minus 119910)) ≳ 119887

119895+119896+ℓObserve that 119887(120582

minus)119904+1

le 119887119904+2 By this (103) 120601 isin S

119904(R119899) and


1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 le 119887

119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119887minus(119904+2)(119895+119896+ℓ)

119887119895+119896

119886119871119902

120593(1199090+119861119895)

≲ [119887(120582minus)119904+1

]minusℓ

119886119871119902

120593(1199090+119861119895)

(104)



and |119860(119895+119896)119911| ≲ (120582minus)(119895+119896)


sup119910isin1199090+119861119895

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816

≲ sup119911isin119861119895+119896

sup|120572|=119904+1

10038161003816100381610038161003816120597120572

120601 (119860119896

(119909 minus 1199090) + 119911)

10038161003816100381610038161003816|119911|119904+1

≲ (120582minus)(119904+1)(119895+119896) sup

119911isin119861119895+119896

[1 + 120588 (119860119896

(119909 minus 1199090) + 119911)]

minus(119904+2)

≲ (120582minus)(119904+1)(119895+119896)min 1 119887minus(119904+2)(119895+119896+ℓ)

(105)


119860119896

(119909 minus 1199090) + 119861

119895+119896sub (119861

119895+119896+ℓ+120590)∁

+ 119861119895+119896

sub (119861119895+119896+ℓ

)∁

(106)


1003816100381610038161003816

le 119887119896

int1199090+119861119895

1003816100381610038161003816119886 (119910)1003816100381610038161003816

10038161003816100381610038161003816120601 (119860

119896

(119909 minus 119910)) minus 119875 (119860119896

(119909 minus 119910))10038161003816100381610038161003816119889119910

≲ 119886119871119902

120593(1199090+119861119895)(120582minus)(119904+1)(119895+119896)

119887119895+119896min 1 119887minus(119904+2)(119895+119896+ℓ)

(107)


le 119887119904+2 we know

that

1003816100381610038161003816119886 lowast 120601119896 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (108)




119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899)

sup119896isinZ

1003816100381610038161003816120601119896 lowast 119886 (119909)1003816100381610038161003816 ≲ [(120582minus)

(119904+1)

119887]minusℓ

119886119871119902

120593(1199090+119861119895) (109)


120601isinS119904(R119899)sup

119896isinZ|119886 lowast 120601119896(119909)|





119867120593119902119904

119860(R119899

) sub 119867120593

119898119860(R119899

) (110)



119898119860(R119899) sub 119867

120593119902119904

119860(R119899)



119898119860(R119899) For each


119898as follows

119891 = 119892119896

+sum

119894

119887119896

119894 (111)

where

Ω119896= 119909 119891

lowast

119898(119909) gt 2

119896

119887119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894

119861119896

119894= 119909

119896

119894+ 119861

ℓ119896

119894

(112)



Ω119896and ℓ119896

119894119894= ℓ



119894119894= 120577


(70) and 119875119896119894119894= 119875


119904(R119899) with


and 119894 119895 let 119875119896+1119894119895


)120577119896

119894


119895



intR119899[119891 (119909) minus 119875

119896+1

119895(119909)] 120577

119896

119894(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

= intR119899119875119896+1

119894119895(119909)119876 (119909) 120577

119896+1

119895(119909) 119889119909

(113)


119896

119894+ 119861

ℓ119896

119894+120590



(i) If 119861119896+1119895

cap 119861119896

119894= 0 then ℓ119896+1

119895le ℓ

119896

119894+ 120590 and 119861119896+1

119895sub 119909

119896

119894+

119861ℓ119896

119894+4120590

(ii) For any 119894 119895 119861119896+1119895

cap 119861119896

119894= 0 le 2119871 where 119871 is as in

(69)




10038161003816100381610038161003816119875119896+1

119894119895(119910) 120577

119896+1

119895(119910)

10038161003816100381610038161003816le 119862

10sup119910isin119880

119891lowast

119898(119910) le 119862

102119896+1

(114)

where 119880 = (119909119896+1

119895+ 119861

ℓ119896+1

119895+4120590+1

) cap (Ω119896+1

)∁


119894119895120577119896+1

119895= 0 where the




119898119860(R119899) Then for any


sum

119896isinZ

120593(Ω1198962119896

120582) le 119862int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909 (115)


119898119860(R119899)



120593(sdot1)(R119899) cap 119867

120593

119898119860(R119899) there



119896


S1015840(R119899) and

supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

forall119896 isin Z 119894 (116)

Ω119896= cup

119894(119909119896

119894+ 119861

ℓ119896

119894+4120590


(119909119896

119894+ 119861

ℓ119896

119894minus2120590

) cap (119909119896

119895+ 119861

ℓ119896

119895minus2120590

) = 0


(118)


10038161003816100381610038161003816119886119896

119894

10038161003816100381610038161003816le 1198622

119896 (119)


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

le 119862intR119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(120)


Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Proof Let 119891 isin 119867120593

119898119860(R119899) cap 119871

119902




lowast

119898

119891 = 119892119896

+ sum119894119887119896



119898119860(R119899)



119892119896


119891 = sum

119896isinZ

(119892119896+1

minus 119892119896

) (121)


119894119887119896

119894) sub Ω

119896and 120593(Ω


then 119892119896


119894120577119896

119894119887119896+1

119895=

120594Ω119896119887119896+1

119895= 119887

119896+1


119892119896+1

minus 119892119896

= (119891 minussum

119895

119887119896+1


119895

119887119896

119895)

= sum

119895

119887119896

119895minussum

119895

119887119896+1

119895+sum

119894

(sum

119895

119875119896+1

119894119895120577119896+1

119895)

= sum

119894

[

[

119887119896

119894minussum

119895

(120577119896

119894119887119896+1

119895minus 119875

119896+1

119894119895120577119896+1

119895)]

]

= sum

119894

119886119896

119894

(122)


119886119896

119894= (119891 minus 119875

119896

119894) 120577119896

119894minussum

119895

[(119891 minus 119875119896+1

119895) 120577119896

119894minus 119875

119896+1

119894119895] 120577119896+1

119895 (123)


119894119895 for all 119876 isin P

119904(R119899) we have

intR119899119886119896

119894(119909)119876 (119909) 119889119909 = 0 (124)


119895= 120594


119886119896

119894= 119891120594

(Ω119896+1)∁120577119896

119894minus 119875

119896

119894120577119896

119894

+sum

119895

119875119896+1

119895120577119896

119894120577119896+1

119895+sum

119895

119875119896+1

119894119895120577119896+1

119895

(125)




that10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)≲ 2

119896

(126)

Recall that 119875119896+1119894119895

= 0 implies 119861119896+1119895

cap 119861119896



sub 119861119896+1

119895sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590


supp 119886119896119894sub 119909

119896

119894+ 119861

ℓ119896

119894+4120590

(127)


119894


sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894minus2120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)

120582)

≲ sum

119896isinZ

120593(Ω1198962119896

120582) ≲ int

R119899120593(119909

119891lowast

119898(119909)

120582) 119889119909

(128)




119895120593(119909

119895+119861

ℓ119895 119905119895120582) le

119888 implies that

inf


119895

120593(119909119895+ 119861

ℓ119895119905119895

120572) le 1

le 119862120582 (129)



(R119899) =

119867120593

119898119860(R119899) = 119867

120593



119867120593infin119904

119860(R119899

) sub 119867120593119902119904

119860(R119899

) sub 119867120593

119860(R119899

) sub 119867120593

119898119860(R119899

) (130)


120593

119898119860(R119899) with 119898 ge


119860(R119899) ≲ 119891

119867120593

119898119860(R119899) which

implies that119867120593119898119860

(R119899) sub 119867120593infin119904

119860(R119899)


119898119860(R119899)cap119871

119902

120593(sdot1)(R119899) by Lemma 38

we obtain

sum

119896isinZ

sum

119894

120593(119909119896

119894+ 119861

ℓ119896

119894+4120590

10038171003817100381710038171003817119886119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

≲ intR119899120593(119909

119891lowast

119898(119909)

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)119889119909

(131)



10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(132)

Let 119891 isin 119867120593



119898119860(R119899) cap 119871

119902

120593(sdot1)(R119899) such

that 119891119896119867120593

119898119860(R119899) le 2

minus119896

119891119867120593

119898119860(R119899) and 119891 = sum

119896isinN 119891119896 in119867120593


119896has an atomic


119894isinN 119889119896

119894in S1015840(R119899) where 119889119896

119894119894isinN are


119896

119894+ 119861

ℓ119896

119894

Since

sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

)

le sum

119896isinN

sum

119894isinN

120593(119909119896

119894+ 119861

ℓ119896

119894

10038171003817100381710038171003817119889119896

119894

10038171003817100381710038171003817119871infin(R119899)

21198961003817100381710038171003817119891119896

1003817100381710038171003817119867120593

119898119860(R119899)

)

≲ sum

119896isinN

1

(2119896)119901≲ 1

(133)


119896

119894isin

119867120593infin119904

119860(R119899) and

10038171003817100381710038171198911003817100381710038171003817119867120593infin119904

119860(R119899)

le Λinfin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119898119860(R119899)

(134)











120593




119860fin(R119899



119899

) is definedby

10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)= inf

Λ119902(119886

119895119896

119895=1

) 119891 =

119896

sum

119895=1

119886119895

119896 isin N 119886119894119896

119894=1are (120593 119902 119904) -atoms

(135)


119860fin(R119899

) is dense in 119867120593119902119904


norm sdot 119867120593119902119904

119860(R119899)







int119870

119892 (119909)120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 lt infin (136)

then


int119870

1003816100381610038161003816119891119898 (119909) minus 119891 (119909)1003816100381610038161003816

120593 (119909 119905)

int119870

120593 (119910 119905) 119889119910119889119909 997888rarr 0


(137)







119867120593

119860(R119899) are


119899

)

(ii) sdot 119867120593infin119904


119867120593




119899

) sub 119867120593

119860(R119899) and

for all 119891 isin 119867120593119902119904

119860fin(R119899

)

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

le10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899) (138)


119860fin(R119899


120593119902119904

119860fin(R119899

)capC(R119899)when 119902 = infin119891

119867120593119902119904

119860fin(R119899)≲ 119891

119867120593

119860(R119899)




119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119860119891119894119899(R119899)) Assume that


120593119902119904

119860fin(R119899

) and 119891119867120593



0isin Z


Ω119896= 119909 isin R

119899

119891lowast

(119909) gt 2119896

(139)


119867120593

119860(R119899) cap 119871

119902



119894119896isinZ119894


119896isinZsum119894 119886119896



(R119899) sub 119867120593119902119904

119860(R119899) and Theorem 40 we

know thatΛ119902(119886

119896

119894) le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

≲ 1 (140)


119910isin119861119891lowast

(119910) for all119909 isin (119861

lowast

)∁

= (1198611198960+4120590


119891lowast

(119909) le 119862 inf119910isin119861

119891lowast

(119910) le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

1003817100381710038171003817119891lowast1003817100381710038171003817119871120593(R119899)

le 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

(141)


119861minus1

119871120593(R119899) Then

Ω119896sub 119861

lowast

= 1198611198960+4120590

forall119896 gt 1198961015840

(142)

Let ℎ = sum119896le1198961015840 sum119894120582119896

119894119886119896


119896gt1198961015840 sum119894120582119896

119894119886119896

119894 where


119896gt1198961015840 Ω119896sub 119861






(119860) we have

[1

|119861|int119861

1003816100381610038161003816119891 (119909)10038161003816100381610038161199021119889119909]

11199021

≲ [1

120593 (119861 1)int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902

120593 (119909 1) 119889119909]

1119902

lt infin

(143)


1 0)-atom



intR119899

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909) 119909

12057210038161003816100381610038161003816119889119909 ≲ sum

119896isinZ

2119896 1003816100381610038161003816Ω119896

1003816100381610038161003816

≲1003817100381710038171003817119891lowast10038171003817100381710038171198711(R119899) lt infin

(144)


119896

119894119886119896

119894here) This



1015840

le 119862120594119861minus1

119871120593(R119899) and 120594119861

minus1

119871120593(R119899) sim 120594

119861lowastminus1

119871120593(R119899) we

obtain

ℎ119871infin(R119899) ≲ sum

119896le1198961015840

2119896

≲ 1198621003817100381710038171003817120594119861

1003817100381710038171003817minus1

119871120593(R119899)

≲1003817100381710038171003817120594119861lowast

1003817100381710038171003817minus1

119871120593(R119899)

(145)





119894119886119896

119894isin 119871

119902



119895Ω

119895+1)

Since supp 119886119896119894sub 119861

ℓ119896

119894+120590

sub Ω119896sub Ω

119895+1for 119896 gt 119895 applying


119895 Ω

119895+1)

sum

119896gt1198961015840

sum

119894

10038161003816100381610038161003816120582119896

119894119886119896

119894(119909)

10038161003816100381610038161003816≲ sum

119896le119895

2119896

≲ 2119895

≲ 119891lowast

(119909) (146)

By 119891 isin 119871119902

120593(119861) sub 119871

119902

120593(119861lowast


120593(119861lowast


119896gt1198961015840 sum119894120582119896

119894119886119896


119871119902

120593(119861lowast


119865119870= (119894 119896) 119896 gt 119896

1015840

|119894| + |119896| le 119870 (147)


(119894119896)isin119865119870120582119896

119894119886119896

119894 Since sum

119896gt1198961015840 sum119894120582119896

119894119886119896

119894converges in

119871119902

120593(119861lowast


119870)120598 is a (120593 119902 119904)-atom Thus 119891 = ℎ + ℓ

119870+ (ℓ minus ℓ

119870) is a


10038171003817100381710038171198911003817100381710038171003817119867120593119902119904

119860fin(R119899)≲ 119862

0+ Λ

119902(119886

119896

119894(119894119896)isin119865119870

) + 120598 ≲ 1 (148)


in 119867120593infin119904

119860fin (R119899

) then 119886119896



119891119871infin(R119899) for any







lt 119896 le 11989610158401015840


120598

1+ ℓ

120598

2with ℓ120598

1= sum

(119894119896)isin1198651120582119896

119894119886119896

119894

and ℓ1205982= sum

(119894119896)isin1198652120582119896

119894119886119896

119894 where

1198651= (119894 119896) 119887

ℓ119896

119894+120590

ge 120575 1198961015840

lt 119896 le 11989610158401015840

(149)

and 1198652= (119894 119896) 119887

ℓ119896

119894+120590

lt 120575 1198961015840

lt 119896 le 11989610158401015840



11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11989610158401015840


119896sub 119861




10158401015840

minus 1198961015840



120598

1+ ℓ

120598

2 By (140) the part ℓ120598

1is


1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)le Λ

infin(119886

119896

119894) ≲

10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

sim 1 (150)



120598

1 Moreover

supp ℓ1205982sub 119861


le 1198621(11989610158401015840

minus 1198961015840




ℓ120598

2= 120582

120598

119886120598 with 120582

120598

= 1198621(11989610158401015840

minus 1198961015840

) 1205981003817100381710038171003817120594119861lowast

1003817100381710038171003817119871120593(R119899)

119886120598

=

1003817100381710038171003817120594119861lowast1003817100381710038171003817minus1

119871120593(R119899)

1198621(11989610158401015840 minus 1198961015840) 120598

(151)

Therefore 119891 = ℎ + ℓ120598

1+ ℓ

120598




119860fin (R119899)≲ ℎ

119867120593infin119904

119860fin (R119899)

+1003817100381710038171003817ℓ120598

1

1003817100381710038171003817119867120593infin119904

119860fin (R119899)+1003817100381710038171003817ℓ120598

2

1003817100381710038171003817119867120593infin119904

119860fin (R119899)

≲ 1

(152)



120593






119891 + 119892120574

B120574le 119891

120574

B120574+ 119892

120574

B120574for all 119891 119892 isin B

120574


119908(R119899) and 119867

119901

119908(R119899 119860)




2(2119870)

minus1





120574

B120574le |120582|

1205741003817100381710038171003817119879(119891)1003817100381710038171003817120574

B120574+ |]|1205741003817100381710038171003817119879(119892)

1003817100381710038171003817120574

B120574(153)


120574-sublinear

Moreover if B120574

= 119871119902




120574-sublinear




120593119902119904

119860119891119894119899(R119899) rarr B

120574is a B

120574-


119878 = sup 119879(119886)B120574 119886 119894119904 119886119899119910 (120593 119902 119904) -119886119905119900119898 lt infin (154)



119878 = sup 119879 (119886)B120574 119886 119894119904 119886119899119910 119888119900119899119905119894119899119906119900119906119904 (120593infin 119904) -119886119905119900119898

lt infin

(155)



119860(R119899) toB

120574


120593119902119904

119860fin(R119899


119895119896

119895=1

and (120593 119902 119904)-atoms 119886119895119896

119895=1

supported in

balls 119909119895+ 119861

ℓ119895119896

119895=1

such that 119891 = sum119896

119895=1120582119895119886119895pointwise By


Λ119902(120582

119895119896

119895=1

)

= inf


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

120582) le 1

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(156)



and 119905 isin (0infin)120593 (119909 119904119905) le 119862

120574119904120574

120593 (119909 119905) (157)



120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574|1205821198950|120574

ge sum119896

119895=1|120582119895|120574

then

119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

ge 120593(1199091198950+ 119861

ℓ1198950

10038171003817100381710038171003817100381710038171205941199091198950+119861ℓ1198950

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) = 1

(158)


119896

sum

119895=1

120593(119909119895+ 119861

ℓ119895

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

119862minus1120574

120574(sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)1120574

)

≳

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

sum119896

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574120593(119909

119895+ 119861

ℓ119895

1003817100381710038171003817100381710038171003817120594119909119895+119861ℓ119895

1003817100381710038171003817100381710038171003817

minus1

119871120593(R119899)

) sim 1

(159)

which implies that

(

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

le 1198621120574

120574Λ119902(120582

119895119896

119895=1

) ≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(160)


10038171003817100381710038171198791198911003817100381710038171003817B120574

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119879 (

119896

sum

119895=1

120582119895119886119895)

1120574100381710038171003817100381710038171003817100381710038171003817100381710038171003817B120574

≲ (

119896

sum

119895=1

10038161003816100381610038161003816120582119895

10038161003816100381610038161003816

120574

)

1120574

≲10038171003817100381710038171198911003817100381710038171003817119867120593

119860(R119899)

(161)

Since119867120593119902119904119860fin(R

119899




120593infin119904

119860fin (R119899

) cap C(R119899) 119879(119891)B120574 ≲ 119891119867120593

119860(R119899) To


119867120593infin119904

119860fin (R119899


120593fin (R119899 119860)


119860fin (R119899



119860(R119899) Actually


119899


119860fin (R119899


120593infin119904

119860fin (R119899


119897 Take 120595 isin S(R119899)


R119899120595(119909) 119889119909 = 1 By (7) it is easy



119896has


119896

120595(119860119896

119909)


120593infin119904

119860fin (R119899


119896) sub 119861

119897+120590for any minus119896 lt 119897




1003817100381710038171003817119891 minus 119891 lowast 1205951198961003817100381710038171003817119871119902



119896 where








Acknowledgments


References



























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























119899







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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