Research ArticleEstimates for Multilinear Commutators of GeneralizedFractional Integral Operators on Weighted Morrey Spaces

Sha He1 Taotao Zheng2 and Xiangxing Tao3

1School of Mathematical Sciences Beijing Normal University and Laboratory of Mathematics and Complex SystemsMinistry of Education Beijing 100875 China2Department of Mathematics Zhejiang University Hangzhou Zhejiang 310027 China3Department of Mathematics Zhejiang University of Science and Technology Hangzhou Zhejiang 310023 China

Correspondence should be addressed to Xiangxing Tao xxtaozusteducn

Received 28 July 2014 Accepted 29 October 2014

Academic Editor Nelson Jose Merentes Dıaz

Copyright copy 2015 Sha He et al This 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 119871 be the infinitesimal generator of an analytic semigroup on 1198712(R119899)with Gaussian kernel bounds and let 119871minus1205722 be the fractional

integrals of 119871 for 0 lt 120572 lt 119899 Assume that = (1198871 1198872 119887

119898) is a finite family of locally integrable functions then the multilinear

commutators generated by and 119871minus1205722 are defined by 119871minus1205722

119891 = [119887

119898 [119887

2 [119887

1 119871minus1205722

]] ]119891 Assume that 119887119895belongs to weighted

BMO space 119895 = 1 2 119898 the authors obtain the boundedness of 119871minus1205722

onweightedMorrey spaces As a special case when 119871 = minusΔ

is the Laplacian operator the authors also obtain the boundedness of the multilinear fractional commutator 119868120572on weightedMorrey

spaces The main results in this paper are substantial improvements and extensions of some known results

1 Introduction and Main Results

Assume that 119871 is a linear operator on 1198712(R119899) which generatesan analytic semigroup 119890

minus119905119871 with a kernel 119901119905(119909 119910) satisfying a

Gaussian upper bound that is

1003816100381610038161003816119901119905 (119909 119910)1003816100381610038161003816 le

119862

1199051198992119890minus119888(|119909minus119910|

2119905) (1)

for 119909 119910 isin R119899 and all 119905 gt 0The property (1) is satisfied by a large amount of differen-

tial operators One can see [1] for details and examplesFor 0 lt 120572 lt 119899 the fractional integral 119871minus1205722 generated by

the operator 119871 is defined by

119871minus1205722

119891 (119909) =1

Γ (1205722)int

infin

0

119890minus119905119871

(119891)119889119905

119905minus1205722+1(119909) (2)

Let = (1198871 1198872 119887

119898) be a finite family of locally

integrable functions then the multilinear commutators gen-erated by 119871minus1205722 and are defined by

119871minus1205722

119891 = [119887

119898 [119887

2 [119887

1 119871minus1205722

]] ] 119891 (3)

where119898 isin Z+Note that if 119871 = minusΔ which is Laplacian onR119899 then 119871

minus1205722

is the classical fractional integral 119868120572

119868120572119891 (119909) =

Γ ((119899 minus 120572) 2)

12058711989922120572Γ (1205722)intR119899

119891 (119910)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572119889119910 (4)

while 119871minus1205722

is the iterated commutator generated by and 119868120572

119868

120572119891 = [119887

119898 [119887

2 [119887

1 119868120572]] ] 119891 (5)

where119898 isin Z+When 119898 = 1 it is easy to see that 119871minus1205722

119891 = [119887 119871

minus1205722]119891

is the commutator generated by 119871minus1205722 and 119887 and when 1198871=

1198872= sdot sdot sdot = 119887

119898 119871minus1205722

is the higher commutator

As we all know if 119887 isin BMO the commutator offractional integral operator [119887 119868

120572] is bounded from 119871

119901(R119899)

to 119871119902(R119899) where 1 lt 119901 lt 119899120572 and 1119902 = 1119901 minus

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 670649 11 pageshttpdxdoiorg1011552015670649

2 Journal of Function Spaces

120572119899 (see [2]) In 2004 Duong and Yan [1] generalized theabove classical result and obtained the (119871119901 119871119902) boundednessof the commutator [119887 119871

minus1205722] under the same conditions

Simultaneously the theory on multilinear integral operatorsand multilinear commutators has attracted much attentionas a rapid developing field in harmonic analysis Mo andLu [3] studied the (119871

119901 119871119902) boundedness of the multilinear

commutators 119871minus1205722

where = (1198871 119887

119898) 119887

119895isin BMO and

119895 = 1 2 119898On the other hand Muckenhoupt andWheeden [4] gave

some definitions of weighted bounded mean oscillation andobtained some equivalent conditions for them

Definition 1 (see [4]) Let 1 le 119901 lt infin and119908 be locally integralin R119899 and let 119908 ge 0 A locally integrable function 119887 is said tobe in BMO

119901(119908) if

119887BMO119901(119908)

= sup119876

(1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876

1003816100381610038161003816

119901

119908 (119909)1minus119901

119889119909)

1119901

le 119862

(6)

where 119887119876

= (1|119876|) int119876119887(119910)119889119910 and the supremum is taken

over all balls 119876 isin R119899We may note that other weighted definitions for the

bounded mean oscillation also have been given by Mucken-houpt and Wheeden in [4]

Definition 2 (see [4]) Let119908 be locally integral inR119899 and119908 ge

0 A locally integrable function 119887 is said to be in BMO(119908) ifthe norm of BMO(119908) sdot

lowast119908satisfies

119887lowast119908

= sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876119908

1003816100381610038161003816 119908 (119909) 119889119909 le 119862 (7)

where 119887119876119908

= (1119908(119876)) int119876119887(119911)119908(119911)119889119911 and the supremum is

taken over all balls 119876 isin R119899The above two definitions cannot contain each other

throughout this paper we will make some investigations onthe basis of Definition 2

Recently Wang [5] obtained some estimates for the com-mutator [119887 119868

120572] on weighted Morrey space (see Definitions

3 and 4) where 119887 isin BMO1(119908) Furthermore Wang and Si

[6] obtained the necessary and sufficient conditions for theboundedness of [119887 119871minus1205722] on weighted Morrey spaces when119887 isin BMO

1(119908)

Motivated by [1 3 5 6] it is natural to raise the followingquestion how to establish corresponding boundedness ofthe multilinear commutator 119871minus1205722

119891 on the weighted Morrey

space where = (1198871 119887

119898) 119887

119895isin BMO(119908)

The question is not motivated only by a mere quest toextend the multilinear commutator 119871minus1205722

119891 from the classical

commutator [119887 119868120572] but rather by their natural appearance in

analysis (see [3])To state the main results we now give some definitions

and notations

A weight is a locally integrable function on R119899 whichtakes values in (0infin) almost everywhere For a weight 119908and a measurable set 119864 we define 119908(119864) = int

119864119908(119909)119889119909 the

Lebesgue measure of 119864 by |119864| and the characteristic functionof 119864 by 120594

119864 For a real number 119901 1 lt 119901 lt infin 1199011015840 is the

conjugate of 119901 that is 1119901 + 11199011015840= 1 The letter 119862 denotes

a positive constant that may vary at each occurrence but isindependent of the essential variable

Definition 3 (see [7]) Let 1 le 119901 lt infin 0 lt 120581 lt 1 let 119908 be aweight then weighted Morrey space is defined by

119871119901120581

(119908) = 119891 isin 119871119901

loc (119908) 1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908)lt infin (8)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

= sup119861

(1

119908 (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

119908 (119909) 119889119909)

1119901

(9)

and the supremum is taken over all balls 119861 in R119899

Definition 4 (see [7]) Let 1 le 119901 lt infin 0 lt 120581 lt 1 let 119906 V beweight then two weights weighted Morrey space are definedby

119871119901120581

(119906 V) = 119891 1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) lt infin (10)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V) = sup

119861

(1

V (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

119906 (119909) 119889119909)

1119901

(11)

and the supremum is taken over all balls 119861 in R119899 If 119906 = Vthen we denote 119871119901120581(119906) for short

Remark 5 (1) If 119908 = 1 120581 = 120582119899 and 0 lt 120582 lt 119899 then119871119901120581

(119908) = 119871119901120582

(R119899) the classical Morrey space(2) If 120581 = 0 then 1198711199010(119908) = 119871

119901(119908) the weighted Lebesgue

space if 119908 = 1 120581 = 0 then 119871119901120581

(119908) = 119871119901(R119899) the Lebesgue

space

Definition 6 (see [8]) A weight function 119908 is in the Muck-enhoupt class 119860

119901with 1 lt 119901 lt infin if for every ball 119861 in R119899

there exists a positive constant 119862 which is independent of 119861such that

(1

|119861|int119861

119908 (119909) 119889119909)(1

|119861|int119861

119908 (119909)minus1(119901minus1)

119889119909)

119901minus1

le 119862 (12)

When 119901 = 1 119908 isin 1198601 if

1

|119861|int119861

119908 (119909) 119889119909 le 119862 ess inf119909isin119861

119908 (119909) (13)

When 119901 = infin119908 isin 119860infin if there exist positive constants 120575 and

119862 such that given a ball 119861 and 119864 is a measurable subset of 119861then

119908 (119864)

119908 (119861)le 119862(

|119864|

|119861|)

120575

(14)

Journal of Function Spaces 3

Definition 7 (see [9]) A weight function 119908 belongs to thereverse Holder class RH

119903 if there exist two constants 119903 gt 1

and 119862 gt 0 such that the following reverse Holder inequality

(1

|119861|int119861

119908 (119909)119903119889119909)

1119903

le 1198621

|119861|int119861

119908 (119909) 119889119909 (15)

holds for every ball 119861 in R119899

It is well known that if 119908 isin 119860119901with 1 le 119901 lt infin then

there exists 119903 gt 1 such that119908 isin RH119903 It follows fromHolderrsquos

inequality that 119908 isin RH119903implies 119908 isin RH

119904for all 1 lt 119904 lt 119903

Moreover if 119908 isin RH119903 119903 gt 1 then we have 119908 isin RH

119903+120576

for some 120576 gt 0 We thus write 119903119908

equiv sup119903 gt 1 119908 isin

RH119903 to denote the critical index of 119908 for the reverse Holder

condition

Definition 8 The Hardy-Littlewood maximal operator 119872 isdefined by

119872119891(119909) = sup119909isin119861

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (16)

Let 119908 be a weight The weighted maximal operator 119872119908is

defined by

119872119908119891 (119909) = sup

119909isin119861

1

119908 (119861)int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119908 (119910) 119889119910 (17)

For 0 lt 120572 lt 119899 119903 ge 1 the fractional maximal operator119872120572119903

isdefined by

119872120572119903119891 (119909) = sup

119909isin119861

(1

|119861|1minus120572119903119899

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119903

119889119910)

1119903

(18)

And the fractional weighted maximal operator 119872120572119903119908

isdefined by

119872120572119903119908

119891 (119909) = sup119909isin119861

(1

119908 (119861)1minus120572119903119899

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119903

119908 (119910) 119889119910)

1119903

(19)

If 120572 = 0 we denote119872119903119908

for short

Definition 9 A family of operators 119860119905 119905 gt 0 is said to

be an ldquoapproximation to identityrdquo if for every 119905 gt 0 119860119905

is represented by the kernel 119886119905(119909 119910) which is a measurable

function defined onR119899timesR119899 in the following sense for every119891 isin 119871

119901(R119899) 119901 ge 1

119860119905119891 (119909) = int

R119899119886119905(119909 119910) 119891 (119910) 119889119910

1003816100381610038161003816119886119905 (119909 119910)1003816100381610038161003816 le ℎ

119905(119909 119910) = 119905

minus1198992119892(

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

119905)

(20)

for (119909 119910) isin R119899 times R119899 119905 gt 0 Here 119892 is a positive boundeddecreasing function satisfying

lim119903rarrinfin

119903119899+120576

119892 (1199032) = 0 (21)

for some 120576 gt 0

Associated with an ldquoapproximation to identityrdquo 119860119905 119905 gt

0 Martell [10] introduced the sharp maximal function asfollows

119872♯

119860119891 (119909) = sup

119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119891 (119910) minus 119860

119905119861

119891 (119910)10038161003816100381610038161003816119889119910 (22)

where 119905119861= 119903

2

119861 119903119861is the radius of the ball 119861 and 119891 isin 119871

119901(R119899)

for some 119901 ge 1Notice that our analytic semigroup 119890

minus119905119871 119905 gt 0 is an

ldquoapproximation to identityrdquo In particular denote

119872♯

119871119891 (119909) = sup

119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119891 (119910) minus 119890

minus119905119861119871119891 (119910)

10038161003816100381610038161003816119889119910 (23)

Next we make some conventions on notation Givenany positive integer 119898 for all 0 le 119895 le 119898 we denote by119862119898

119895the family of all finite subsets 120590 = 120590

1 1205902 120590

119895 of

1 2 119898 of different elements and for any 120590 isin 119862119898

119895 let

1205901015840= 1 2 119898 120590 Let = (119887

1 1198872 119887

119898) then for any

120590 = 1205901 1205902 120590

119895 isin 119862

119898

119895 we denote

120590= (119887

1205901

1198871205902

119887120590119895

)119887120590(119909) = prod

120590119895isin120590119887120590119895

(119909) and 120590BMO(119908) = prod

120590119895isin120590119887120590119895

BMO(119908) andBMO(119908) = prod

120590119895isin12119898

119887120590119895

BMO(119908) = prod119898

119895=1119887119895BMO(119908)

In this paper our main results are stated as follows

Theorem 10 Assume the condition (1) holds Let 0 lt 120572 lt 1198991 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899 0 le 120581 lt 119901119902 119908119902119901 isin 119860

1

and 119903119908gt (1 minus 120581)(119901119902 minus 120581) where 119903

119908denotes the critical index

of 119908 for the reverse Holder condition If 119887119895isin 119861119872119874(119908) 119895 =

1 2 119898 then

100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(24)

Theorem 11 Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899119908119902119901

isin 1198601 and 119903

119908gt 119902119901 where 119903

119908denotes the critical index

of 119908 for the reverse Holder condition If 119887119895isin 119861119872119874(119908) 119895 =

1 2 119898 then

100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902(119908119902119901)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901(119908)

(25)

Moreover if 119871 = minusΔ is the Laplacian then

100381710038171003817100381710038171003817119868

120572119891100381710038171003817100381710038171003817119871119902(119908119902119901)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901(119908)

(26)

Remark 12 We note that our results extend some results in[1 3] To be specific if we take 119898 = 1 119908 = 1 and 120581 = 0

in Theorem 10 it is easy to see that our conclusion is themain result of Duong and Yan [1] If we only take 119908 = 1 and120581 = 0 in Theorem 10 our result contains the correspondingconclusion of [3]

The remaining part of this paper will be organized asfollows In Section 2 we will give some known results andprove some requisite lemmas Section 3 is devoted to provingthe theorems of this paper

4 Journal of Function Spaces

2 Requisite Lemmas

In this section we will prove some lemmas and state someknown results about weights and weighted Morrey space

Lemma 13 (see [11]) Let 119904 gt 1 1 le 119901 lt infin and 119860119904

119901= 119908

119908119904isin 119860

119901 Then

119860119904

119901= 119860

1+(119901minus1)119904cap 119877119867

119904 (27)

In particular 1198601199041= 119860

1cap 119877119867

119904

Lemma 14 (see [10]) Assume that the semigroup 119890minus119905119871 has a

kernel 119901119905(119909 119910) which satisfies the upper bound (1) Take 120582 gt 0

119891 isin 1198711

0(R119899) (the set of functions in 119871

1(R119899) with bounded

support) and a ball 1198610such that there exists 119909

0isin 119861

0with

119872119891(1199090) le 120582 Then for every 119908 isin 119860

infin 0 lt 120578 lt 1 we can

find 120574 gt 0 (independent of 120582 1198610 119891 119909

0) and constants 119862 119903 gt 0

(which only depend on 119908) such that

119908(119909 isin 1198610 119872119891 (119909) gt 119860120582119872

♯

119871119891 (119909) le 120574120582) le 119862120578

119903119908 (119861

0)

(28)

where 119860 gt 1 is a fixed constant which depends only on 119899

As a result using the above good-120582 inequality togetherwith the standard arguments we have the following estimates

For every 119891 isin 119871119901120581

(119906 V) 1 lt 119901 lt infin 0 le 120581 lt 1 if119906 V isin 119860

infin then1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119906V) le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119906V)

(29)

In particular when 119906 = V = 119908 119908 isin 119860infin we have

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119908)le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119908)

(30)

Lemma 15 (Kolmogorovrsquos inequality see [9 page 455]) Let0 lt 119903 lt 119897 lt infin for 119891 ge 0 define 119891

119871119897infin = sup

119905gt0119905|119909 isin

R119899 |119891(119909)| gt 119905|1119897119873

119897119903(119891) = sup

119864(119891120594

119864119903120594

119864ℎ) and 1ℎ =

1119903 minus 1119897 then

10038171003817100381710038171198911003817100381710038171003817119871119897infin

le 119873119897119903(119891) le (

119897

119897 minus 119903)

1119903

10038171003817100381710038171198911003817100381710038171003817119871119897infin

(31)

Lemma 16 (see [4]) Let119908 isin 119860infinThen the norm of119861119872119874(119908)

sdot lowast119908

is equivalent to the norm of 119861119872119874(R119899) sdot lowast that is if

119887 is a locally integrable function then

119887lowast119908 = sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876119908

1003816100381610038161003816 119908 (119909) 119889119909 le 119862 (32)

is equivalent to

119887lowast= sup

119876

1

|119876|int119876

1003816100381610038161003816119887 (119909) minus 119887119876

1003816100381610038161003816 119889119909 le 119862 (33)

Lemma 17 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901minus

120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581)

then10038171003817100381710038171198721205721

1198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (34)

It also holds for 119868120572

Lemma 18 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 1 lt 119903 lt 119901

119903119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171198721199031199081198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908) (35)

Lemma 19 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 0 lt 120581 lt 119901119902 and 119908 isin 119860infin Then for 1 lt 119903 lt 119901

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (36)

Remark 20 It is easy to see that Lemmas 17 18 and 19 stillhold for 120581 = 0

Lemma 21 Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899and 119908119902119901 isin 119860

1 if 0 le 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (37)

Proof Since semigroup 119890minus119905119871 has a kernel 119901119905(119909 119910) that satisfies

the upper bound (1) it is easy to see that for 119909 isin R119899119871minus1205722

119891(119909) le 119868120572(|119891|)(119909) From the boundedness of 119868

120572on

weighted Morrey space (see Lemma 17) we get

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le1003817100381710038171003817119868120572119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(38)

Remark 22 Since 119868120572is of weak (1 119899(119899 minus 120572)) type from the

above proof we can obtain that 119871minus1205722 is of weak (1 119899(119899 minus120572))

type

Lemma 23 (see [1]) Assume the semigroup 119890minus119905119871 has a kernel

119901119905(119909 119910) which satisfies the upper bound (1) Then for 0 lt 120572 lt

119899 the differential operator 119871minus1205722 minus 119890minus119905119871

119871minus1205722 has an associated

kernel 120572119905(119909 119910) which satisfies

120572119905

(119909 119910) le119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

119905

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2 (39)

Lemma 24 Assume the semigroup 119890minus119905119871 has a kernel 119901

119905(119909 119910)

which satisfies the upper bound (1) 119887 isin 119861119872119874(119908) and119908 isin 1198601

Then for 119891 isin 119871119901(R119899) 119901 gt 1 120590 isin 119862

119898

119895(119895 = 1 2 119898) 1 lt

120591 lt infin and

sup119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817119861119872119874(119908)119872120591119908

119891 (119909)

(40)

where 119905119861= 119903

2

119861and 119903

119861is the radius of 119861

Journal of Function Spaces 5

Proof For any 119891 isin 119871119901(R119899) 119909 isin 119861 we have

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le1

|119861|int119861

intR119899

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

+1

|119861|int119861

infin

sum

119896=1

int2119896+11198612119896119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times1003816100381610038161003816(119887 (119911) minus 119887

119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

= 119868 + 119868119868

(41)

Noticing that 119910 isin 119861 119911 isin 2119861 from (1) we get10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862119905

minus1198992

119861le 119862 |2119861|

minus1 (42)

Thus

119868 le119862

|2119861|int2119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

prod

120590119895isin120590

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(43)

For simplicity we only consider the case of 119895 = 2 We alsowant to point out that althoughwe state our results on the caseof 119895 = 2 all results are valid on the multilinear case (119895 gt 2)

without any essential difference and difficulty in the proof Soit follows that

119868 le119862

|2119861|int2119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681+ 119868

2+ 119868

3+ 119868

4

(44)

We split 1198681as follows

1198681le

119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times 1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

1003816100381610038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986811+ 119868

12+ 119868

13+ 119868

14

(45)

We now consider the four terms respectively Choose1205911 1205912 120591 119904 gt 1 satisfing 1120591

1+ 1120591

2+ 1120591 + 1119904 = 1 According

to Holderrsquos inequality and 119908 isin 1198601 we have

11986811

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

11205911

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

112059121003816100381610038161003816119891 (119911)

1003816100381610038161003816

times 119908 (119911)1120591

119908 (119911)minus1+1119904

119889119911

le119862

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1205911

119908 (119911) 119889119911)

11205911

times (int2119861

1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1205912

119908 (119911) 119889119911)

11205912

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minus119904+1

119889119911)

1119904

le119862

|2119861|

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119908119872120591119908

119891 (119909)119908 (2119861)1minus1119904

119908 (119909)minus1+1119904

|2119861|1119904

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1119904

119908 (119909)minus1+1119904

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(46)

In the above inequalities we use the fact that if 119908 isin 1198601

then 119908 isin 119860infin Thus the norm of BMO(119908) is equivalent to

the norm of BMO(R119899) (see Lemma 16)For 119868

12 we first estimate the term that contains (119887

1205902

)2119861

In fact it follows from the John-Nirenberg lemma that thereexist 119862

1gt 0 and 119862

2gt 0 such that for any ball 119861 and 120572 gt 0

10038161003816100381610038161003816119911 isin 2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572

10038161003816100381610038161003816le 119862

1 |2119861| 119890minus11986221205721198871205902lowast

(47)

6 Journal of Function Spaces

since 119887 isin BMO(R119899) Using the definition of 119860infin we get

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572)

le 119862119908 (2119861) 119890minus1198622120572120575119887

1205902lowast

(48)

for some 120575 gt 0 We can see that (48) yields

int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

= int

infin

0

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572) 119889120572

le 119862119908 (2119861)int

infin

0

119890minus1198622120572120575119887

1205902lowast119889120572

= 119862119908 (2119861)100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(49)

Thus

100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

le1

119908 (2119861)int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

le119862

119908 (2119861)119908 (2119861)

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast= 119862

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(50)

On the basis of (50) we now estimate 11986812 For the above 120591

select 119906 V such that 1119906 + 1V + 1120591 = 1 by virtue of Holderrsquosinequality and 119908 isin 119860

1 we have

11986812

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times 119908 (119911)1119906 1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119908 (119911)1120591

119908 (119911)minus1+1V

119889119911

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

119906

119908 (119911) 119889119911)

1119906

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minusV+1

119889119911)

1V

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)1

|2119861|119908 (2119861)

1119906+1120591119908 (119909)

minus1+1V|2119861|

1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1V

119908 (119909)minus1+1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(51)

Analogous to the estimate of 11986812 we also have 119868

13le

119862120590lowast119872120591119908

119891(119909)

As for 11986814 taking advantage of Holderrsquos inequality and

(50) we get

11986814

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119908 (119911)

1120591119908 (119911)

11205911015840minus1

119889119911

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|(int

2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

times (int2119861

119908 (119911)1minus1205911015840

119889119911)

11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|119872120591119908

119891 (119909)119908 (2119861)1120591

119908 (119909)11205911015840minus1

|2119861|11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1120591

119908 (119909)11205911015840minus1

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(52)

Collecting the estimates of 11986811 11986812 11986813 and 119868

14 it is easy to see

that

1198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (53)

For term 1198682 using the fact that |119887

2119861minus 119887

119861| le 119862119887

lowast(see

[12]) we now get

1198682le100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

119862

|2119861|int2119861

(1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816)

times1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986821+ 119868

22

(54)

By some estimates similar to those used in the estimate for11986812 we conclude that

11986821

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (55)

For 11986822 using the same method as in dealing with 119868

14 we get

11986822

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (56)

Therefore

1198682le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (57)

Analogously

1198683le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (58)

Journal of Function Spaces 7

An argument similar to that used in the estimate for 11986814leads

to

1198684le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (59)

Hence

119868 le 1198681+ 119868

2+ 119868

3+ 119868

4

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909)

(60)

Next we will consider the second term 119868119868 in (41) For any119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 it is easy to get that |119910 minus 119911| ge 2

119896minus1119903119861

and

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

(61)

Thus

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

int2119896+1119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)119861

10038161003816100381610038161003816sdot sdot sdot

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(62)

For simplicity we also consider the case of 119895 = 2

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681198681+ 119868119868

2+ 119868119868

3+ 119868119868

4

(63)

For 1198681198681 similar to the estimate of 119868

1 we have

1198681198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(64)

Noticing thatwe could use similarmethods of the estimates of1198682 1198683 and 119868

4in the estimates of 119868119868

2 119868119868

3 and 119868119868

4 respectively

From this together with the fact that |1198872119895+1119861minus 119887

119861| le 2

119899(119895 +

1)119887lowast(see [12]) it is easy to get

1198681198682+ 119868119868

3+ 119868119868

4le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (65)

Therefore

119868119868 le 119862120590lowast119872120591119908

119891 (119909) (66)

According to the estimates of 119868 and 119868119868 the lemma has beenproved

Now we will establish a lemma which plays an importantrole in the proof of Theorem 10

Lemma 25 Let 0 lt 120572 lt 119899 119908 isin 1198601 and 119887 isin 119861119872119874(119908) then

for all 119903 gt 1 120591 gt 1 and 119909 isin R119899 we have

119872♯

119871(119871

minus1205722

119891) (119909)

le 119862 1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

119908 (119909)minus120572119899

119872120572119903119908

119891 (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(67)

Proof For any given 119909 isin R119899 take a ball 119861 = 119861(1199090 119903119861) which

contains 119909 For 119891 isin 119871119901(R119899) let 119891

1= 119891120594

2119861 119891

2= 119891 minus 119891

1

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

120572119899 (see [2]) In 2004 Duong and Yan [1] generalized theabove classical result and obtained the (119871119901 119871119902) boundednessof the commutator [119887 119871

minus1205722] under the same conditions

Simultaneously the theory on multilinear integral operatorsand multilinear commutators has attracted much attentionas a rapid developing field in harmonic analysis Mo andLu [3] studied the (119871

119901 119871119902) boundedness of the multilinear

commutators 119871minus1205722

where = (1198871 119887

119898) 119887

119895isin BMO and

119895 = 1 2 119898On the other hand Muckenhoupt andWheeden [4] gave

some definitions of weighted bounded mean oscillation andobtained some equivalent conditions for them

Definition 1 (see [4]) Let 1 le 119901 lt infin and119908 be locally integralin R119899 and let 119908 ge 0 A locally integrable function 119887 is said tobe in BMO

119901(119908) if

119887BMO119901(119908)

= sup119876

(1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876

1003816100381610038161003816

119901

119908 (119909)1minus119901

119889119909)

1119901

le 119862

(6)

where 119887119876

= (1|119876|) int119876119887(119910)119889119910 and the supremum is taken

over all balls 119876 isin R119899We may note that other weighted definitions for the

bounded mean oscillation also have been given by Mucken-houpt and Wheeden in [4]

Definition 2 (see [4]) Let119908 be locally integral inR119899 and119908 ge

0 A locally integrable function 119887 is said to be in BMO(119908) ifthe norm of BMO(119908) sdot

lowast119908satisfies

119887lowast119908

= sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876119908

1003816100381610038161003816 119908 (119909) 119889119909 le 119862 (7)

where 119887119876119908

= (1119908(119876)) int119876119887(119911)119908(119911)119889119911 and the supremum is

taken over all balls 119876 isin R119899The above two definitions cannot contain each other

throughout this paper we will make some investigations onthe basis of Definition 2

Recently Wang [5] obtained some estimates for the com-mutator [119887 119868

120572] on weighted Morrey space (see Definitions

3 and 4) where 119887 isin BMO1(119908) Furthermore Wang and Si

[6] obtained the necessary and sufficient conditions for theboundedness of [119887 119871minus1205722] on weighted Morrey spaces when119887 isin BMO

1(119908)

Motivated by [1 3 5 6] it is natural to raise the followingquestion how to establish corresponding boundedness ofthe multilinear commutator 119871minus1205722

119891 on the weighted Morrey

space where = (1198871 119887

119898) 119887

119895isin BMO(119908)

The question is not motivated only by a mere quest toextend the multilinear commutator 119871minus1205722

119891 from the classical

commutator [119887 119868120572] but rather by their natural appearance in

analysis (see [3])To state the main results we now give some definitions

and notations

A weight is a locally integrable function on R119899 whichtakes values in (0infin) almost everywhere For a weight 119908and a measurable set 119864 we define 119908(119864) = int

119864119908(119909)119889119909 the

Lebesgue measure of 119864 by |119864| and the characteristic functionof 119864 by 120594

119864 For a real number 119901 1 lt 119901 lt infin 1199011015840 is the

conjugate of 119901 that is 1119901 + 11199011015840= 1 The letter 119862 denotes

a positive constant that may vary at each occurrence but isindependent of the essential variable

Definition 3 (see [7]) Let 1 le 119901 lt infin 0 lt 120581 lt 1 let 119908 be aweight then weighted Morrey space is defined by

119871119901120581

(119908) = 119891 isin 119871119901

loc (119908) 1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908)lt infin (8)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

= sup119861

(1

119908 (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

119908 (119909) 119889119909)

1119901

(9)

and the supremum is taken over all balls 119861 in R119899

Definition 4 (see [7]) Let 1 le 119901 lt infin 0 lt 120581 lt 1 let 119906 V beweight then two weights weighted Morrey space are definedby

119871119901120581

(119906 V) = 119891 1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) lt infin (10)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119906V) = sup

119861

(1

V (119861)120581int119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

119906 (119909) 119889119909)

1119901

(11)

and the supremum is taken over all balls 119861 in R119899 If 119906 = Vthen we denote 119871119901120581(119906) for short

Remark 5 (1) If 119908 = 1 120581 = 120582119899 and 0 lt 120582 lt 119899 then119871119901120581

(119908) = 119871119901120582

(R119899) the classical Morrey space(2) If 120581 = 0 then 1198711199010(119908) = 119871

119901(119908) the weighted Lebesgue

space if 119908 = 1 120581 = 0 then 119871119901120581

(119908) = 119871119901(R119899) the Lebesgue

space

Definition 6 (see [8]) A weight function 119908 is in the Muck-enhoupt class 119860

119901with 1 lt 119901 lt infin if for every ball 119861 in R119899

there exists a positive constant 119862 which is independent of 119861such that

(1

|119861|int119861

119908 (119909) 119889119909)(1

|119861|int119861

119908 (119909)minus1(119901minus1)

119889119909)

119901minus1

le 119862 (12)

When 119901 = 1 119908 isin 1198601 if

1

|119861|int119861

119908 (119909) 119889119909 le 119862 ess inf119909isin119861

119908 (119909) (13)

When 119901 = infin119908 isin 119860infin if there exist positive constants 120575 and

119862 such that given a ball 119861 and 119864 is a measurable subset of 119861then

119908 (119864)

119908 (119861)le 119862(

|119864|

|119861|)

120575

(14)

Journal of Function Spaces 3

Definition 7 (see [9]) A weight function 119908 belongs to thereverse Holder class RH

119903 if there exist two constants 119903 gt 1

and 119862 gt 0 such that the following reverse Holder inequality

(1

|119861|int119861

119908 (119909)119903119889119909)

1119903

le 1198621

|119861|int119861

119908 (119909) 119889119909 (15)

holds for every ball 119861 in R119899

It is well known that if 119908 isin 119860119901with 1 le 119901 lt infin then

there exists 119903 gt 1 such that119908 isin RH119903 It follows fromHolderrsquos

inequality that 119908 isin RH119903implies 119908 isin RH

119904for all 1 lt 119904 lt 119903

Moreover if 119908 isin RH119903 119903 gt 1 then we have 119908 isin RH

119903+120576

for some 120576 gt 0 We thus write 119903119908

equiv sup119903 gt 1 119908 isin

RH119903 to denote the critical index of 119908 for the reverse Holder

condition

Definition 8 The Hardy-Littlewood maximal operator 119872 isdefined by

119872119891(119909) = sup119909isin119861

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (16)

Let 119908 be a weight The weighted maximal operator 119872119908is

defined by

119872119908119891 (119909) = sup

119909isin119861

1

119908 (119861)int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119908 (119910) 119889119910 (17)

For 0 lt 120572 lt 119899 119903 ge 1 the fractional maximal operator119872120572119903

isdefined by

119872120572119903119891 (119909) = sup

119909isin119861

(1

|119861|1minus120572119903119899

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119903

119889119910)

1119903

(18)

And the fractional weighted maximal operator 119872120572119903119908

isdefined by

119872120572119903119908

119891 (119909) = sup119909isin119861

(1

119908 (119861)1minus120572119903119899

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119903

119908 (119910) 119889119910)

1119903

(19)

If 120572 = 0 we denote119872119903119908

for short

Definition 9 A family of operators 119860119905 119905 gt 0 is said to

be an ldquoapproximation to identityrdquo if for every 119905 gt 0 119860119905

is represented by the kernel 119886119905(119909 119910) which is a measurable

function defined onR119899timesR119899 in the following sense for every119891 isin 119871

119901(R119899) 119901 ge 1

119860119905119891 (119909) = int

R119899119886119905(119909 119910) 119891 (119910) 119889119910

1003816100381610038161003816119886119905 (119909 119910)1003816100381610038161003816 le ℎ

119905(119909 119910) = 119905

minus1198992119892(

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

119905)

(20)

for (119909 119910) isin R119899 times R119899 119905 gt 0 Here 119892 is a positive boundeddecreasing function satisfying

lim119903rarrinfin

119903119899+120576

119892 (1199032) = 0 (21)

for some 120576 gt 0

Associated with an ldquoapproximation to identityrdquo 119860119905 119905 gt

0 Martell [10] introduced the sharp maximal function asfollows

119872♯

119860119891 (119909) = sup

119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119891 (119910) minus 119860

119905119861

119891 (119910)10038161003816100381610038161003816119889119910 (22)

where 119905119861= 119903

2

119861 119903119861is the radius of the ball 119861 and 119891 isin 119871

119901(R119899)

for some 119901 ge 1Notice that our analytic semigroup 119890

minus119905119871 119905 gt 0 is an

ldquoapproximation to identityrdquo In particular denote

119872♯

119871119891 (119909) = sup

119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119891 (119910) minus 119890

minus119905119861119871119891 (119910)

10038161003816100381610038161003816119889119910 (23)

Next we make some conventions on notation Givenany positive integer 119898 for all 0 le 119895 le 119898 we denote by119862119898

119895the family of all finite subsets 120590 = 120590

1 1205902 120590

119895 of

1 2 119898 of different elements and for any 120590 isin 119862119898

119895 let

1205901015840= 1 2 119898 120590 Let = (119887

1 1198872 119887

119898) then for any

120590 = 1205901 1205902 120590

119895 isin 119862

119898

119895 we denote

120590= (119887

1205901

1198871205902

119887120590119895

)119887120590(119909) = prod

120590119895isin120590119887120590119895

(119909) and 120590BMO(119908) = prod

120590119895isin120590119887120590119895

BMO(119908) andBMO(119908) = prod

120590119895isin12119898

119887120590119895

BMO(119908) = prod119898

119895=1119887119895BMO(119908)

In this paper our main results are stated as follows

Theorem 10 Assume the condition (1) holds Let 0 lt 120572 lt 1198991 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899 0 le 120581 lt 119901119902 119908119902119901 isin 119860

1

and 119903119908gt (1 minus 120581)(119901119902 minus 120581) where 119903

119908denotes the critical index

of 119908 for the reverse Holder condition If 119887119895isin 119861119872119874(119908) 119895 =

1 2 119898 then

100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(24)

Theorem 11 Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899119908119902119901

isin 1198601 and 119903

119908gt 119902119901 where 119903

119908denotes the critical index

of 119908 for the reverse Holder condition If 119887119895isin 119861119872119874(119908) 119895 =

1 2 119898 then

100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902(119908119902119901)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901(119908)

(25)

Moreover if 119871 = minusΔ is the Laplacian then

100381710038171003817100381710038171003817119868

120572119891100381710038171003817100381710038171003817119871119902(119908119902119901)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901(119908)

(26)

Remark 12 We note that our results extend some results in[1 3] To be specific if we take 119898 = 1 119908 = 1 and 120581 = 0

in Theorem 10 it is easy to see that our conclusion is themain result of Duong and Yan [1] If we only take 119908 = 1 and120581 = 0 in Theorem 10 our result contains the correspondingconclusion of [3]

The remaining part of this paper will be organized asfollows In Section 2 we will give some known results andprove some requisite lemmas Section 3 is devoted to provingthe theorems of this paper

4 Journal of Function Spaces

2 Requisite Lemmas

In this section we will prove some lemmas and state someknown results about weights and weighted Morrey space

Lemma 13 (see [11]) Let 119904 gt 1 1 le 119901 lt infin and 119860119904

119901= 119908

119908119904isin 119860

119901 Then

119860119904

119901= 119860

1+(119901minus1)119904cap 119877119867

119904 (27)

In particular 1198601199041= 119860

1cap 119877119867

119904

Lemma 14 (see [10]) Assume that the semigroup 119890minus119905119871 has a

kernel 119901119905(119909 119910) which satisfies the upper bound (1) Take 120582 gt 0

119891 isin 1198711

0(R119899) (the set of functions in 119871

1(R119899) with bounded

support) and a ball 1198610such that there exists 119909

0isin 119861

0with

119872119891(1199090) le 120582 Then for every 119908 isin 119860

infin 0 lt 120578 lt 1 we can

find 120574 gt 0 (independent of 120582 1198610 119891 119909

0) and constants 119862 119903 gt 0

(which only depend on 119908) such that

119908(119909 isin 1198610 119872119891 (119909) gt 119860120582119872

♯

119871119891 (119909) le 120574120582) le 119862120578

119903119908 (119861

0)

(28)

where 119860 gt 1 is a fixed constant which depends only on 119899

As a result using the above good-120582 inequality togetherwith the standard arguments we have the following estimates

For every 119891 isin 119871119901120581

(119906 V) 1 lt 119901 lt infin 0 le 120581 lt 1 if119906 V isin 119860

infin then1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119906V) le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119906V)

(29)

In particular when 119906 = V = 119908 119908 isin 119860infin we have

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119908)le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119908)

(30)

Lemma 15 (Kolmogorovrsquos inequality see [9 page 455]) Let0 lt 119903 lt 119897 lt infin for 119891 ge 0 define 119891

119871119897infin = sup

119905gt0119905|119909 isin

R119899 |119891(119909)| gt 119905|1119897119873

119897119903(119891) = sup

119864(119891120594

119864119903120594

119864ℎ) and 1ℎ =

1119903 minus 1119897 then

10038171003817100381710038171198911003817100381710038171003817119871119897infin

le 119873119897119903(119891) le (

119897

119897 minus 119903)

1119903

10038171003817100381710038171198911003817100381710038171003817119871119897infin

(31)

Lemma 16 (see [4]) Let119908 isin 119860infinThen the norm of119861119872119874(119908)

sdot lowast119908

is equivalent to the norm of 119861119872119874(R119899) sdot lowast that is if

119887 is a locally integrable function then

119887lowast119908 = sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876119908

1003816100381610038161003816 119908 (119909) 119889119909 le 119862 (32)

is equivalent to

119887lowast= sup

119876

1

|119876|int119876

1003816100381610038161003816119887 (119909) minus 119887119876

1003816100381610038161003816 119889119909 le 119862 (33)

Lemma 17 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901minus

120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581)

then10038171003817100381710038171198721205721

1198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (34)

It also holds for 119868120572

Lemma 18 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 1 lt 119903 lt 119901

119903119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171198721199031199081198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908) (35)

Lemma 19 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 0 lt 120581 lt 119901119902 and 119908 isin 119860infin Then for 1 lt 119903 lt 119901

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (36)

Remark 20 It is easy to see that Lemmas 17 18 and 19 stillhold for 120581 = 0

Lemma 21 Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899and 119908119902119901 isin 119860

1 if 0 le 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (37)

Proof Since semigroup 119890minus119905119871 has a kernel 119901119905(119909 119910) that satisfies

the upper bound (1) it is easy to see that for 119909 isin R119899119871minus1205722

119891(119909) le 119868120572(|119891|)(119909) From the boundedness of 119868

120572on

weighted Morrey space (see Lemma 17) we get

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le1003817100381710038171003817119868120572119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(38)

Remark 22 Since 119868120572is of weak (1 119899(119899 minus 120572)) type from the

above proof we can obtain that 119871minus1205722 is of weak (1 119899(119899 minus120572))

type

Lemma 23 (see [1]) Assume the semigroup 119890minus119905119871 has a kernel

119901119905(119909 119910) which satisfies the upper bound (1) Then for 0 lt 120572 lt

119899 the differential operator 119871minus1205722 minus 119890minus119905119871

119871minus1205722 has an associated

kernel 120572119905(119909 119910) which satisfies

120572119905

(119909 119910) le119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

119905

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2 (39)

Lemma 24 Assume the semigroup 119890minus119905119871 has a kernel 119901

119905(119909 119910)

which satisfies the upper bound (1) 119887 isin 119861119872119874(119908) and119908 isin 1198601

Then for 119891 isin 119871119901(R119899) 119901 gt 1 120590 isin 119862

119898

119895(119895 = 1 2 119898) 1 lt

120591 lt infin and

sup119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817119861119872119874(119908)119872120591119908

119891 (119909)

(40)

where 119905119861= 119903

2

119861and 119903

119861is the radius of 119861

Journal of Function Spaces 5

Proof For any 119891 isin 119871119901(R119899) 119909 isin 119861 we have

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le1

|119861|int119861

intR119899

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

+1

|119861|int119861

infin

sum

119896=1

int2119896+11198612119896119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times1003816100381610038161003816(119887 (119911) minus 119887

119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

= 119868 + 119868119868

(41)

Noticing that 119910 isin 119861 119911 isin 2119861 from (1) we get10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862119905

minus1198992

119861le 119862 |2119861|

minus1 (42)

Thus

119868 le119862

|2119861|int2119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

prod

120590119895isin120590

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(43)

For simplicity we only consider the case of 119895 = 2 We alsowant to point out that althoughwe state our results on the caseof 119895 = 2 all results are valid on the multilinear case (119895 gt 2)

without any essential difference and difficulty in the proof Soit follows that

119868 le119862

|2119861|int2119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681+ 119868

2+ 119868

3+ 119868

4

(44)

We split 1198681as follows

1198681le

119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times 1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

1003816100381610038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986811+ 119868

12+ 119868

13+ 119868

14

(45)

We now consider the four terms respectively Choose1205911 1205912 120591 119904 gt 1 satisfing 1120591

1+ 1120591

2+ 1120591 + 1119904 = 1 According

to Holderrsquos inequality and 119908 isin 1198601 we have

11986811

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

11205911

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

112059121003816100381610038161003816119891 (119911)

1003816100381610038161003816

times 119908 (119911)1120591

119908 (119911)minus1+1119904

119889119911

le119862

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1205911

119908 (119911) 119889119911)

11205911

times (int2119861

1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1205912

119908 (119911) 119889119911)

11205912

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minus119904+1

119889119911)

1119904

le119862

|2119861|

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119908119872120591119908

119891 (119909)119908 (2119861)1minus1119904

119908 (119909)minus1+1119904

|2119861|1119904

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1119904

119908 (119909)minus1+1119904

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(46)

In the above inequalities we use the fact that if 119908 isin 1198601

then 119908 isin 119860infin Thus the norm of BMO(119908) is equivalent to

the norm of BMO(R119899) (see Lemma 16)For 119868

12 we first estimate the term that contains (119887

1205902

)2119861

In fact it follows from the John-Nirenberg lemma that thereexist 119862

1gt 0 and 119862

2gt 0 such that for any ball 119861 and 120572 gt 0

10038161003816100381610038161003816119911 isin 2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572

10038161003816100381610038161003816le 119862

1 |2119861| 119890minus11986221205721198871205902lowast

(47)

6 Journal of Function Spaces

since 119887 isin BMO(R119899) Using the definition of 119860infin we get

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572)

le 119862119908 (2119861) 119890minus1198622120572120575119887

1205902lowast

(48)

for some 120575 gt 0 We can see that (48) yields

int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

= int

infin

0

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572) 119889120572

le 119862119908 (2119861)int

infin

0

119890minus1198622120572120575119887

1205902lowast119889120572

= 119862119908 (2119861)100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(49)

Thus

100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

le1

119908 (2119861)int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

le119862

119908 (2119861)119908 (2119861)

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast= 119862

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(50)

On the basis of (50) we now estimate 11986812 For the above 120591

select 119906 V such that 1119906 + 1V + 1120591 = 1 by virtue of Holderrsquosinequality and 119908 isin 119860

1 we have

11986812

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times 119908 (119911)1119906 1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119908 (119911)1120591

119908 (119911)minus1+1V

119889119911

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

119906

119908 (119911) 119889119911)

1119906

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minusV+1

119889119911)

1V

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)1

|2119861|119908 (2119861)

1119906+1120591119908 (119909)

minus1+1V|2119861|

1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1V

119908 (119909)minus1+1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(51)

Analogous to the estimate of 11986812 we also have 119868

13le

119862120590lowast119872120591119908

119891(119909)

As for 11986814 taking advantage of Holderrsquos inequality and

(50) we get

11986814

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119908 (119911)

1120591119908 (119911)

11205911015840minus1

119889119911

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|(int

2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

times (int2119861

119908 (119911)1minus1205911015840

119889119911)

11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|119872120591119908

119891 (119909)119908 (2119861)1120591

119908 (119909)11205911015840minus1

|2119861|11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1120591

119908 (119909)11205911015840minus1

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(52)

Collecting the estimates of 11986811 11986812 11986813 and 119868

14 it is easy to see

that

1198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (53)

For term 1198682 using the fact that |119887

2119861minus 119887

119861| le 119862119887

lowast(see

[12]) we now get

1198682le100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

119862

|2119861|int2119861

(1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816)

times1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986821+ 119868

22

(54)

By some estimates similar to those used in the estimate for11986812 we conclude that

11986821

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (55)

For 11986822 using the same method as in dealing with 119868

14 we get

11986822

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (56)

Therefore

1198682le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (57)

Analogously

1198683le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (58)

Journal of Function Spaces 7

An argument similar to that used in the estimate for 11986814leads

to

1198684le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (59)

Hence

119868 le 1198681+ 119868

2+ 119868

3+ 119868

4

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909)

(60)

Next we will consider the second term 119868119868 in (41) For any119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 it is easy to get that |119910 minus 119911| ge 2

119896minus1119903119861

and

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

(61)

Thus

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

int2119896+1119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)119861

10038161003816100381610038161003816sdot sdot sdot

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(62)

For simplicity we also consider the case of 119895 = 2

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681198681+ 119868119868

2+ 119868119868

3+ 119868119868

4

(63)

For 1198681198681 similar to the estimate of 119868

1 we have

1198681198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(64)

Noticing thatwe could use similarmethods of the estimates of1198682 1198683 and 119868

4in the estimates of 119868119868

2 119868119868

3 and 119868119868

4 respectively

From this together with the fact that |1198872119895+1119861minus 119887

119861| le 2

119899(119895 +

1)119887lowast(see [12]) it is easy to get

1198681198682+ 119868119868

3+ 119868119868

4le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (65)

Therefore

119868119868 le 119862120590lowast119872120591119908

119891 (119909) (66)

According to the estimates of 119868 and 119868119868 the lemma has beenproved

Now we will establish a lemma which plays an importantrole in the proof of Theorem 10

Lemma 25 Let 0 lt 120572 lt 119899 119908 isin 1198601 and 119887 isin 119861119872119874(119908) then

for all 119903 gt 1 120591 gt 1 and 119909 isin R119899 we have

119872♯

119871(119871

minus1205722

119891) (119909)

le 119862 1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

119908 (119909)minus120572119899

119872120572119903119908

119891 (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(67)

Proof For any given 119909 isin R119899 take a ball 119861 = 119861(1199090 119903119861) which

contains 119909 For 119891 isin 119871119901(R119899) let 119891

1= 119891120594

2119861 119891

2= 119891 minus 119891

1

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

Definition 7 (see [9]) A weight function 119908 belongs to thereverse Holder class RH

119903 if there exist two constants 119903 gt 1

and 119862 gt 0 such that the following reverse Holder inequality

(1

|119861|int119861

119908 (119909)119903119889119909)

1119903

le 1198621

|119861|int119861

119908 (119909) 119889119909 (15)

holds for every ball 119861 in R119899

It is well known that if 119908 isin 119860119901with 1 le 119901 lt infin then

there exists 119903 gt 1 such that119908 isin RH119903 It follows fromHolderrsquos

inequality that 119908 isin RH119903implies 119908 isin RH

119904for all 1 lt 119904 lt 119903

Moreover if 119908 isin RH119903 119903 gt 1 then we have 119908 isin RH

119903+120576

for some 120576 gt 0 We thus write 119903119908

equiv sup119903 gt 1 119908 isin

RH119903 to denote the critical index of 119908 for the reverse Holder

condition

Definition 8 The Hardy-Littlewood maximal operator 119872 isdefined by

119872119891(119909) = sup119909isin119861

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 (16)

Let 119908 be a weight The weighted maximal operator 119872119908is

defined by

119872119908119891 (119909) = sup

119909isin119861

1

119908 (119861)int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119908 (119910) 119889119910 (17)

For 0 lt 120572 lt 119899 119903 ge 1 the fractional maximal operator119872120572119903

isdefined by

119872120572119903119891 (119909) = sup

119909isin119861

(1

|119861|1minus120572119903119899

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119903

119889119910)

1119903

(18)

And the fractional weighted maximal operator 119872120572119903119908

isdefined by

119872120572119903119908

119891 (119909) = sup119909isin119861

(1

119908 (119861)1minus120572119903119899

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119903

119908 (119910) 119889119910)

1119903

(19)

If 120572 = 0 we denote119872119903119908

for short

Definition 9 A family of operators 119860119905 119905 gt 0 is said to

be an ldquoapproximation to identityrdquo if for every 119905 gt 0 119860119905

is represented by the kernel 119886119905(119909 119910) which is a measurable

function defined onR119899timesR119899 in the following sense for every119891 isin 119871

119901(R119899) 119901 ge 1

119860119905119891 (119909) = int

R119899119886119905(119909 119910) 119891 (119910) 119889119910

1003816100381610038161003816119886119905 (119909 119910)1003816100381610038161003816 le ℎ

119905(119909 119910) = 119905

minus1198992119892(

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

119905)

(20)

for (119909 119910) isin R119899 times R119899 119905 gt 0 Here 119892 is a positive boundeddecreasing function satisfying

lim119903rarrinfin

119903119899+120576

119892 (1199032) = 0 (21)

for some 120576 gt 0

Associated with an ldquoapproximation to identityrdquo 119860119905 119905 gt

0 Martell [10] introduced the sharp maximal function asfollows

119872♯

119860119891 (119909) = sup

119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119891 (119910) minus 119860

119905119861

119891 (119910)10038161003816100381610038161003816119889119910 (22)

where 119905119861= 119903

2

119861 119903119861is the radius of the ball 119861 and 119891 isin 119871

119901(R119899)

for some 119901 ge 1Notice that our analytic semigroup 119890

minus119905119871 119905 gt 0 is an

ldquoapproximation to identityrdquo In particular denote

119872♯

119871119891 (119909) = sup

119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119891 (119910) minus 119890

minus119905119861119871119891 (119910)

10038161003816100381610038161003816119889119910 (23)

Next we make some conventions on notation Givenany positive integer 119898 for all 0 le 119895 le 119898 we denote by119862119898

119895the family of all finite subsets 120590 = 120590

1 1205902 120590

119895 of

1 2 119898 of different elements and for any 120590 isin 119862119898

119895 let

1205901015840= 1 2 119898 120590 Let = (119887

1 1198872 119887

119898) then for any

120590 = 1205901 1205902 120590

119895 isin 119862

119898

119895 we denote

120590= (119887

1205901

1198871205902

119887120590119895

)119887120590(119909) = prod

120590119895isin120590119887120590119895

(119909) and 120590BMO(119908) = prod

120590119895isin120590119887120590119895

BMO(119908) andBMO(119908) = prod

120590119895isin12119898

119887120590119895

BMO(119908) = prod119898

119895=1119887119895BMO(119908)

In this paper our main results are stated as follows

Theorem 10 Assume the condition (1) holds Let 0 lt 120572 lt 1198991 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899 0 le 120581 lt 119901119902 119908119902119901 isin 119860

1

and 119903119908gt (1 minus 120581)(119901119902 minus 120581) where 119903

119908denotes the critical index

of 119908 for the reverse Holder condition If 119887119895isin 119861119872119874(119908) 119895 =

1 2 119898 then

100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(24)

Theorem 11 Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899119908119902119901

isin 1198601 and 119903

119908gt 119902119901 where 119903

119908denotes the critical index

of 119908 for the reverse Holder condition If 119887119895isin 119861119872119874(119908) 119895 =

1 2 119898 then

100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902(119908119902119901)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901(119908)

(25)

Moreover if 119871 = minusΔ is the Laplacian then

100381710038171003817100381710038171003817119868

120572119891100381710038171003817100381710038171003817119871119902(119908119902119901)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817119861119872119874(119908)

10038171003817100381710038171198911003817100381710038171003817119871119901(119908)

(26)

Remark 12 We note that our results extend some results in[1 3] To be specific if we take 119898 = 1 119908 = 1 and 120581 = 0

in Theorem 10 it is easy to see that our conclusion is themain result of Duong and Yan [1] If we only take 119908 = 1 and120581 = 0 in Theorem 10 our result contains the correspondingconclusion of [3]

The remaining part of this paper will be organized asfollows In Section 2 we will give some known results andprove some requisite lemmas Section 3 is devoted to provingthe theorems of this paper

4 Journal of Function Spaces

2 Requisite Lemmas

In this section we will prove some lemmas and state someknown results about weights and weighted Morrey space

Lemma 13 (see [11]) Let 119904 gt 1 1 le 119901 lt infin and 119860119904

119901= 119908

119908119904isin 119860

119901 Then

119860119904

119901= 119860

1+(119901minus1)119904cap 119877119867

119904 (27)

In particular 1198601199041= 119860

1cap 119877119867

119904

Lemma 14 (see [10]) Assume that the semigroup 119890minus119905119871 has a

kernel 119901119905(119909 119910) which satisfies the upper bound (1) Take 120582 gt 0

119891 isin 1198711

0(R119899) (the set of functions in 119871

1(R119899) with bounded

support) and a ball 1198610such that there exists 119909

0isin 119861

0with

119872119891(1199090) le 120582 Then for every 119908 isin 119860

infin 0 lt 120578 lt 1 we can

find 120574 gt 0 (independent of 120582 1198610 119891 119909

0) and constants 119862 119903 gt 0

(which only depend on 119908) such that

119908(119909 isin 1198610 119872119891 (119909) gt 119860120582119872

♯

119871119891 (119909) le 120574120582) le 119862120578

119903119908 (119861

0)

(28)

where 119860 gt 1 is a fixed constant which depends only on 119899

As a result using the above good-120582 inequality togetherwith the standard arguments we have the following estimates

For every 119891 isin 119871119901120581

(119906 V) 1 lt 119901 lt infin 0 le 120581 lt 1 if119906 V isin 119860

infin then1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119906V) le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119906V)

(29)

In particular when 119906 = V = 119908 119908 isin 119860infin we have

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119908)le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119908)

(30)

Lemma 15 (Kolmogorovrsquos inequality see [9 page 455]) Let0 lt 119903 lt 119897 lt infin for 119891 ge 0 define 119891

119871119897infin = sup

119905gt0119905|119909 isin

R119899 |119891(119909)| gt 119905|1119897119873

119897119903(119891) = sup

119864(119891120594

119864119903120594

119864ℎ) and 1ℎ =

1119903 minus 1119897 then

10038171003817100381710038171198911003817100381710038171003817119871119897infin

le 119873119897119903(119891) le (

119897

119897 minus 119903)

1119903

10038171003817100381710038171198911003817100381710038171003817119871119897infin

(31)

Lemma 16 (see [4]) Let119908 isin 119860infinThen the norm of119861119872119874(119908)

sdot lowast119908

is equivalent to the norm of 119861119872119874(R119899) sdot lowast that is if

119887 is a locally integrable function then

119887lowast119908 = sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876119908

1003816100381610038161003816 119908 (119909) 119889119909 le 119862 (32)

is equivalent to

119887lowast= sup

119876

1

|119876|int119876

1003816100381610038161003816119887 (119909) minus 119887119876

1003816100381610038161003816 119889119909 le 119862 (33)

Lemma 17 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901minus

120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581)

then10038171003817100381710038171198721205721

1198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (34)

It also holds for 119868120572

Lemma 18 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 1 lt 119903 lt 119901

119903119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171198721199031199081198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908) (35)

Lemma 19 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 0 lt 120581 lt 119901119902 and 119908 isin 119860infin Then for 1 lt 119903 lt 119901

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (36)

Remark 20 It is easy to see that Lemmas 17 18 and 19 stillhold for 120581 = 0

Lemma 21 Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899and 119908119902119901 isin 119860

1 if 0 le 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (37)

Proof Since semigroup 119890minus119905119871 has a kernel 119901119905(119909 119910) that satisfies

the upper bound (1) it is easy to see that for 119909 isin R119899119871minus1205722

119891(119909) le 119868120572(|119891|)(119909) From the boundedness of 119868

120572on

weighted Morrey space (see Lemma 17) we get

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le1003817100381710038171003817119868120572119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(38)

Remark 22 Since 119868120572is of weak (1 119899(119899 minus 120572)) type from the

above proof we can obtain that 119871minus1205722 is of weak (1 119899(119899 minus120572))

type

Lemma 23 (see [1]) Assume the semigroup 119890minus119905119871 has a kernel

119901119905(119909 119910) which satisfies the upper bound (1) Then for 0 lt 120572 lt

119899 the differential operator 119871minus1205722 minus 119890minus119905119871

119871minus1205722 has an associated

kernel 120572119905(119909 119910) which satisfies

120572119905

(119909 119910) le119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

119905

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2 (39)

Lemma 24 Assume the semigroup 119890minus119905119871 has a kernel 119901

119905(119909 119910)

which satisfies the upper bound (1) 119887 isin 119861119872119874(119908) and119908 isin 1198601

Then for 119891 isin 119871119901(R119899) 119901 gt 1 120590 isin 119862

119898

119895(119895 = 1 2 119898) 1 lt

120591 lt infin and

sup119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817119861119872119874(119908)119872120591119908

119891 (119909)

(40)

where 119905119861= 119903

2

119861and 119903

119861is the radius of 119861

Journal of Function Spaces 5

Proof For any 119891 isin 119871119901(R119899) 119909 isin 119861 we have

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le1

|119861|int119861

intR119899

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

+1

|119861|int119861

infin

sum

119896=1

int2119896+11198612119896119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times1003816100381610038161003816(119887 (119911) minus 119887

119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

= 119868 + 119868119868

(41)

Noticing that 119910 isin 119861 119911 isin 2119861 from (1) we get10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862119905

minus1198992

119861le 119862 |2119861|

minus1 (42)

Thus

119868 le119862

|2119861|int2119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

prod

120590119895isin120590

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(43)

For simplicity we only consider the case of 119895 = 2 We alsowant to point out that althoughwe state our results on the caseof 119895 = 2 all results are valid on the multilinear case (119895 gt 2)

without any essential difference and difficulty in the proof Soit follows that

119868 le119862

|2119861|int2119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681+ 119868

2+ 119868

3+ 119868

4

(44)

We split 1198681as follows

1198681le

119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times 1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

1003816100381610038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986811+ 119868

12+ 119868

13+ 119868

14

(45)

We now consider the four terms respectively Choose1205911 1205912 120591 119904 gt 1 satisfing 1120591

1+ 1120591

2+ 1120591 + 1119904 = 1 According

to Holderrsquos inequality and 119908 isin 1198601 we have

11986811

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

11205911

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

112059121003816100381610038161003816119891 (119911)

1003816100381610038161003816

times 119908 (119911)1120591

119908 (119911)minus1+1119904

119889119911

le119862

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1205911

119908 (119911) 119889119911)

11205911

times (int2119861

1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1205912

119908 (119911) 119889119911)

11205912

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minus119904+1

119889119911)

1119904

le119862

|2119861|

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119908119872120591119908

119891 (119909)119908 (2119861)1minus1119904

119908 (119909)minus1+1119904

|2119861|1119904

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1119904

119908 (119909)minus1+1119904

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(46)

In the above inequalities we use the fact that if 119908 isin 1198601

then 119908 isin 119860infin Thus the norm of BMO(119908) is equivalent to

the norm of BMO(R119899) (see Lemma 16)For 119868

12 we first estimate the term that contains (119887

1205902

)2119861

In fact it follows from the John-Nirenberg lemma that thereexist 119862

1gt 0 and 119862

2gt 0 such that for any ball 119861 and 120572 gt 0

10038161003816100381610038161003816119911 isin 2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572

10038161003816100381610038161003816le 119862

1 |2119861| 119890minus11986221205721198871205902lowast

(47)

6 Journal of Function Spaces

since 119887 isin BMO(R119899) Using the definition of 119860infin we get

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572)

le 119862119908 (2119861) 119890minus1198622120572120575119887

1205902lowast

(48)

for some 120575 gt 0 We can see that (48) yields

int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

= int

infin

0

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572) 119889120572

le 119862119908 (2119861)int

infin

0

119890minus1198622120572120575119887

1205902lowast119889120572

= 119862119908 (2119861)100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(49)

Thus

100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

le1

119908 (2119861)int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

le119862

119908 (2119861)119908 (2119861)

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast= 119862

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(50)

On the basis of (50) we now estimate 11986812 For the above 120591

select 119906 V such that 1119906 + 1V + 1120591 = 1 by virtue of Holderrsquosinequality and 119908 isin 119860

1 we have

11986812

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times 119908 (119911)1119906 1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119908 (119911)1120591

119908 (119911)minus1+1V

119889119911

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

119906

119908 (119911) 119889119911)

1119906

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minusV+1

119889119911)

1V

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)1

|2119861|119908 (2119861)

1119906+1120591119908 (119909)

minus1+1V|2119861|

1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1V

119908 (119909)minus1+1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(51)

Analogous to the estimate of 11986812 we also have 119868

13le

119862120590lowast119872120591119908

119891(119909)

As for 11986814 taking advantage of Holderrsquos inequality and

(50) we get

11986814

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119908 (119911)

1120591119908 (119911)

11205911015840minus1

119889119911

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|(int

2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

times (int2119861

119908 (119911)1minus1205911015840

119889119911)

11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|119872120591119908

119891 (119909)119908 (2119861)1120591

119908 (119909)11205911015840minus1

|2119861|11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1120591

119908 (119909)11205911015840minus1

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(52)

Collecting the estimates of 11986811 11986812 11986813 and 119868

14 it is easy to see

that

1198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (53)

For term 1198682 using the fact that |119887

2119861minus 119887

119861| le 119862119887

lowast(see

[12]) we now get

1198682le100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

119862

|2119861|int2119861

(1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816)

times1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986821+ 119868

22

(54)

By some estimates similar to those used in the estimate for11986812 we conclude that

11986821

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (55)

For 11986822 using the same method as in dealing with 119868

14 we get

11986822

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (56)

Therefore

1198682le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (57)

Analogously

1198683le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (58)

Journal of Function Spaces 7

An argument similar to that used in the estimate for 11986814leads

to

1198684le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (59)

Hence

119868 le 1198681+ 119868

2+ 119868

3+ 119868

4

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909)

(60)

Next we will consider the second term 119868119868 in (41) For any119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 it is easy to get that |119910 minus 119911| ge 2

119896minus1119903119861

and

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

(61)

Thus

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

int2119896+1119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)119861

10038161003816100381610038161003816sdot sdot sdot

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(62)

For simplicity we also consider the case of 119895 = 2

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681198681+ 119868119868

2+ 119868119868

3+ 119868119868

4

(63)

For 1198681198681 similar to the estimate of 119868

1 we have

1198681198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(64)

Noticing thatwe could use similarmethods of the estimates of1198682 1198683 and 119868

4in the estimates of 119868119868

2 119868119868

3 and 119868119868

4 respectively

From this together with the fact that |1198872119895+1119861minus 119887

119861| le 2

119899(119895 +

1)119887lowast(see [12]) it is easy to get

1198681198682+ 119868119868

3+ 119868119868

4le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (65)

Therefore

119868119868 le 119862120590lowast119872120591119908

119891 (119909) (66)

According to the estimates of 119868 and 119868119868 the lemma has beenproved

Now we will establish a lemma which plays an importantrole in the proof of Theorem 10

Lemma 25 Let 0 lt 120572 lt 119899 119908 isin 1198601 and 119887 isin 119861119872119874(119908) then

for all 119903 gt 1 120591 gt 1 and 119909 isin R119899 we have

119872♯

119871(119871

minus1205722

119891) (119909)

le 119862 1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

119908 (119909)minus120572119899

119872120572119903119908

119891 (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(67)

Proof For any given 119909 isin R119899 take a ball 119861 = 119861(1199090 119903119861) which

contains 119909 For 119891 isin 119871119901(R119899) let 119891

1= 119891120594

2119861 119891

2= 119891 minus 119891

1

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

2 Requisite Lemmas

In this section we will prove some lemmas and state someknown results about weights and weighted Morrey space

Lemma 13 (see [11]) Let 119904 gt 1 1 le 119901 lt infin and 119860119904

119901= 119908

119908119904isin 119860

119901 Then

119860119904

119901= 119860

1+(119901minus1)119904cap 119877119867

119904 (27)

In particular 1198601199041= 119860

1cap 119877119867

119904

Lemma 14 (see [10]) Assume that the semigroup 119890minus119905119871 has a

kernel 119901119905(119909 119910) which satisfies the upper bound (1) Take 120582 gt 0

119891 isin 1198711

0(R119899) (the set of functions in 119871

1(R119899) with bounded

support) and a ball 1198610such that there exists 119909

0isin 119861

0with

119872119891(1199090) le 120582 Then for every 119908 isin 119860

infin 0 lt 120578 lt 1 we can

find 120574 gt 0 (independent of 120582 1198610 119891 119909

0) and constants 119862 119903 gt 0

(which only depend on 119908) such that

119908(119909 isin 1198610 119872119891 (119909) gt 119860120582119872

♯

119871119891 (119909) le 120574120582) le 119862120578

119903119908 (119861

0)

(28)

where 119860 gt 1 is a fixed constant which depends only on 119899

As a result using the above good-120582 inequality togetherwith the standard arguments we have the following estimates

For every 119891 isin 119871119901120581

(119906 V) 1 lt 119901 lt infin 0 le 120581 lt 1 if119906 V isin 119860

infin then1003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119906V) le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119906V) le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119906V)

(29)

In particular when 119906 = V = 119908 119908 isin 119860infin we have

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le1003817100381710038171003817119872119891

1003817100381710038171003817119871119901120581(119908)le10038171003817100381710038171003817119872♯

11987111989110038171003817100381710038171003817119871119901120581(119908)

(30)

Lemma 15 (Kolmogorovrsquos inequality see [9 page 455]) Let0 lt 119903 lt 119897 lt infin for 119891 ge 0 define 119891

119871119897infin = sup

119905gt0119905|119909 isin

R119899 |119891(119909)| gt 119905|1119897119873

119897119903(119891) = sup

119864(119891120594

119864119903120594

119864ℎ) and 1ℎ =

1119903 minus 1119897 then

10038171003817100381710038171198911003817100381710038171003817119871119897infin

le 119873119897119903(119891) le (

119897

119897 minus 119903)

1119903

10038171003817100381710038171198911003817100381710038171003817119871119897infin

(31)

Lemma 16 (see [4]) Let119908 isin 119860infinThen the norm of119861119872119874(119908)

sdot lowast119908

is equivalent to the norm of 119861119872119874(R119899) sdot lowast that is if

119887 is a locally integrable function then

119887lowast119908 = sup119876

1

119908 (119876)int119876

1003816100381610038161003816119887 (119909) minus 119887119876119908

1003816100381610038161003816 119908 (119909) 119889119909 le 119862 (32)

is equivalent to

119887lowast= sup

119876

1

|119876|int119876

1003816100381610038161003816119887 (119909) minus 119887119876

1003816100381610038161003816 119889119909 le 119862 (33)

Lemma 17 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901minus

120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581)

then10038171003817100381710038171198721205721

1198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (34)

It also holds for 119868120572

Lemma 18 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 and 119908119902119901

isin 1198601 if 0 lt 120581 lt 119901119902 1 lt 119903 lt 119901

119903119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171198721199031199081198911003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908) (35)

Lemma 19 (see [5]) Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 =

1119901 minus 120572119899 0 lt 120581 lt 119901119902 and 119908 isin 119860infin Then for 1 lt 119903 lt 119901

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (36)

Remark 20 It is easy to see that Lemmas 17 18 and 19 stillhold for 120581 = 0

Lemma 21 Let 0 lt 120572 lt 119899 1 lt 119901 lt 119899120572 1119902 = 1119901 minus 120572119899and 119908119902119901 isin 119860

1 if 0 le 120581 lt 119901119902 119903

119908gt (1 minus 120581)(119901119902 minus 120581) then

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(119908) (37)

Proof Since semigroup 119890minus119905119871 has a kernel 119901119905(119909 119910) that satisfies

the upper bound (1) it is easy to see that for 119909 isin R119899119871minus1205722

119891(119909) le 119868120572(|119891|)(119909) From the boundedness of 119868

120572on

weighted Morrey space (see Lemma 17) we get

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le1003817100381710038171003817119868120572119891

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(38)

Remark 22 Since 119868120572is of weak (1 119899(119899 minus 120572)) type from the

above proof we can obtain that 119871minus1205722 is of weak (1 119899(119899 minus120572))

type

Lemma 23 (see [1]) Assume the semigroup 119890minus119905119871 has a kernel

119901119905(119909 119910) which satisfies the upper bound (1) Then for 0 lt 120572 lt

119899 the differential operator 119871minus1205722 minus 119890minus119905119871

119871minus1205722 has an associated

kernel 120572119905(119909 119910) which satisfies

120572119905

(119909 119910) le119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899minus120572

119905

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2 (39)

Lemma 24 Assume the semigroup 119890minus119905119871 has a kernel 119901

119905(119909 119910)

which satisfies the upper bound (1) 119887 isin 119861119872119874(119908) and119908 isin 1198601

Then for 119891 isin 119871119901(R119899) 119901 gt 1 120590 isin 119862

119898

119895(119895 = 1 2 119898) 1 lt

120591 lt infin and

sup119909isin119861

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817119861119872119874(119908)119872120591119908

119891 (119909)

(40)

where 119905119861= 119903

2

119861and 119903

119861is the radius of 119861

Journal of Function Spaces 5

Proof For any 119891 isin 119871119901(R119899) 119909 isin 119861 we have

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le1

|119861|int119861

intR119899

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

+1

|119861|int119861

infin

sum

119896=1

int2119896+11198612119896119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times1003816100381610038161003816(119887 (119911) minus 119887

119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

= 119868 + 119868119868

(41)

Noticing that 119910 isin 119861 119911 isin 2119861 from (1) we get10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862119905

minus1198992

119861le 119862 |2119861|

minus1 (42)

Thus

119868 le119862

|2119861|int2119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

prod

120590119895isin120590

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(43)

For simplicity we only consider the case of 119895 = 2 We alsowant to point out that althoughwe state our results on the caseof 119895 = 2 all results are valid on the multilinear case (119895 gt 2)

without any essential difference and difficulty in the proof Soit follows that

119868 le119862

|2119861|int2119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681+ 119868

2+ 119868

3+ 119868

4

(44)

We split 1198681as follows

1198681le

119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times 1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

1003816100381610038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986811+ 119868

12+ 119868

13+ 119868

14

(45)

We now consider the four terms respectively Choose1205911 1205912 120591 119904 gt 1 satisfing 1120591

1+ 1120591

2+ 1120591 + 1119904 = 1 According

to Holderrsquos inequality and 119908 isin 1198601 we have

11986811

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

11205911

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

112059121003816100381610038161003816119891 (119911)

1003816100381610038161003816

times 119908 (119911)1120591

119908 (119911)minus1+1119904

119889119911

le119862

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1205911

119908 (119911) 119889119911)

11205911

times (int2119861

1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1205912

119908 (119911) 119889119911)

11205912

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minus119904+1

119889119911)

1119904

le119862

|2119861|

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119908119872120591119908

119891 (119909)119908 (2119861)1minus1119904

119908 (119909)minus1+1119904

|2119861|1119904

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1119904

119908 (119909)minus1+1119904

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(46)

In the above inequalities we use the fact that if 119908 isin 1198601

then 119908 isin 119860infin Thus the norm of BMO(119908) is equivalent to

the norm of BMO(R119899) (see Lemma 16)For 119868

12 we first estimate the term that contains (119887

1205902

)2119861

In fact it follows from the John-Nirenberg lemma that thereexist 119862

1gt 0 and 119862

2gt 0 such that for any ball 119861 and 120572 gt 0

10038161003816100381610038161003816119911 isin 2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572

10038161003816100381610038161003816le 119862

1 |2119861| 119890minus11986221205721198871205902lowast

(47)

6 Journal of Function Spaces

since 119887 isin BMO(R119899) Using the definition of 119860infin we get

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572)

le 119862119908 (2119861) 119890minus1198622120572120575119887

1205902lowast

(48)

for some 120575 gt 0 We can see that (48) yields

int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

= int

infin

0

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572) 119889120572

le 119862119908 (2119861)int

infin

0

119890minus1198622120572120575119887

1205902lowast119889120572

= 119862119908 (2119861)100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(49)

Thus

100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

le1

119908 (2119861)int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

le119862

119908 (2119861)119908 (2119861)

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast= 119862

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(50)

On the basis of (50) we now estimate 11986812 For the above 120591

select 119906 V such that 1119906 + 1V + 1120591 = 1 by virtue of Holderrsquosinequality and 119908 isin 119860

1 we have

11986812

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times 119908 (119911)1119906 1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119908 (119911)1120591

119908 (119911)minus1+1V

119889119911

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

119906

119908 (119911) 119889119911)

1119906

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minusV+1

119889119911)

1V

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)1

|2119861|119908 (2119861)

1119906+1120591119908 (119909)

minus1+1V|2119861|

1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1V

119908 (119909)minus1+1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(51)

Analogous to the estimate of 11986812 we also have 119868

13le

119862120590lowast119872120591119908

119891(119909)

As for 11986814 taking advantage of Holderrsquos inequality and

(50) we get

11986814

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119908 (119911)

1120591119908 (119911)

11205911015840minus1

119889119911

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|(int

2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

times (int2119861

119908 (119911)1minus1205911015840

119889119911)

11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|119872120591119908

119891 (119909)119908 (2119861)1120591

119908 (119909)11205911015840minus1

|2119861|11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1120591

119908 (119909)11205911015840minus1

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(52)

Collecting the estimates of 11986811 11986812 11986813 and 119868

14 it is easy to see

that

1198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (53)

For term 1198682 using the fact that |119887

2119861minus 119887

119861| le 119862119887

lowast(see

[12]) we now get

1198682le100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

119862

|2119861|int2119861

(1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816)

times1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986821+ 119868

22

(54)

By some estimates similar to those used in the estimate for11986812 we conclude that

11986821

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (55)

For 11986822 using the same method as in dealing with 119868

14 we get

11986822

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (56)

Therefore

1198682le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (57)

Analogously

1198683le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (58)

Journal of Function Spaces 7

An argument similar to that used in the estimate for 11986814leads

to

1198684le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (59)

Hence

119868 le 1198681+ 119868

2+ 119868

3+ 119868

4

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909)

(60)

Next we will consider the second term 119868119868 in (41) For any119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 it is easy to get that |119910 minus 119911| ge 2

119896minus1119903119861

and

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

(61)

Thus

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

int2119896+1119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)119861

10038161003816100381610038161003816sdot sdot sdot

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(62)

For simplicity we also consider the case of 119895 = 2

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681198681+ 119868119868

2+ 119868119868

3+ 119868119868

4

(63)

For 1198681198681 similar to the estimate of 119868

1 we have

1198681198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(64)

Noticing thatwe could use similarmethods of the estimates of1198682 1198683 and 119868

4in the estimates of 119868119868

2 119868119868

3 and 119868119868

4 respectively

From this together with the fact that |1198872119895+1119861minus 119887

119861| le 2

119899(119895 +

1)119887lowast(see [12]) it is easy to get

1198681198682+ 119868119868

3+ 119868119868

4le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (65)

Therefore

119868119868 le 119862120590lowast119872120591119908

119891 (119909) (66)

According to the estimates of 119868 and 119868119868 the lemma has beenproved

Now we will establish a lemma which plays an importantrole in the proof of Theorem 10

Lemma 25 Let 0 lt 120572 lt 119899 119908 isin 1198601 and 119887 isin 119861119872119874(119908) then

for all 119903 gt 1 120591 gt 1 and 119909 isin R119899 we have

119872♯

119871(119871

minus1205722

119891) (119909)

le 119862 1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

119908 (119909)minus120572119899

119872120572119903119908

119891 (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(67)

Proof For any given 119909 isin R119899 take a ball 119861 = 119861(1199090 119903119861) which

contains 119909 For 119891 isin 119871119901(R119899) let 119891

1= 119891120594

2119861 119891

2= 119891 minus 119891

1

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

Proof For any 119891 isin 119871119901(R119899) 119909 isin 119861 we have

1

|119861|int119861

10038161003816100381610038161003816119890minus119905119861119871((119887 minus 119887

119861)120590119891) (119910)

10038161003816100381610038161003816119889119910

le1

|119861|int119861

intR119899

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

1003816100381610038161003816(119887 (119911) minus 119887119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

+1

|119861|int119861

infin

sum

119896=1

int2119896+11198612119896119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times1003816100381610038161003816(119887 (119911) minus 119887

119861)120590119891 (119911)

1003816100381610038161003816 119889119911 119889119910

= 119868 + 119868119868

(41)

Noticing that 119910 isin 119861 119911 isin 2119861 from (1) we get10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862119905

minus1198992

119861le 119862 |2119861|

minus1 (42)

Thus

119868 le119862

|2119861|int2119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

prod

120590119895isin120590

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(43)

For simplicity we only consider the case of 119895 = 2 We alsowant to point out that althoughwe state our results on the caseof 119895 = 2 all results are valid on the multilinear case (119895 gt 2)

without any essential difference and difficulty in the proof Soit follows that

119868 le119862

|2119861|int2119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816+10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

le119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

10038161003816100381610038161003816(1198871205901

)2119861

minus (1198871205901

)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198871205902

)2119861

minus (1198871205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681+ 119868

2+ 119868

3+ 119868

4

(44)

We split 1198681as follows

1198681le

119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times 1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

1003816100381610038161003816100381610038161003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+119862

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

times100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986811+ 119868

12+ 119868

13+ 119868

14

(45)

We now consider the four terms respectively Choose1205911 1205912 120591 119904 gt 1 satisfing 1120591

1+ 1120591

2+ 1120591 + 1119904 = 1 According

to Holderrsquos inequality and 119908 isin 1198601 we have

11986811

=119862

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

11205911

times1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816119908 (119911)

112059121003816100381610038161003816119891 (119911)

1003816100381610038161003816

times 119908 (119911)1120591

119908 (119911)minus1+1119904

119889119911

le119862

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1205911

119908 (119911) 119889119911)

11205911

times (int2119861

1003816100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861119908

100381610038161003816100381610038161003816

1205912

119908 (119911) 119889119911)

11205912

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minus119904+1

119889119911)

1119904

le119862

|2119861|

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119908119872120591119908

119891 (119909)119908 (2119861)1minus1119904

119908 (119909)minus1+1119904

|2119861|1119904

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1119904

119908 (119909)minus1+1119904

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(46)

In the above inequalities we use the fact that if 119908 isin 1198601

then 119908 isin 119860infin Thus the norm of BMO(119908) is equivalent to

the norm of BMO(R119899) (see Lemma 16)For 119868

12 we first estimate the term that contains (119887

1205902

)2119861

In fact it follows from the John-Nirenberg lemma that thereexist 119862

1gt 0 and 119862

2gt 0 such that for any ball 119861 and 120572 gt 0

10038161003816100381610038161003816119911 isin 2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572

10038161003816100381610038161003816le 119862

1 |2119861| 119890minus11986221205721198871205902lowast

(47)

6 Journal of Function Spaces

since 119887 isin BMO(R119899) Using the definition of 119860infin we get

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572)

le 119862119908 (2119861) 119890minus1198622120572120575119887

1205902lowast

(48)

for some 120575 gt 0 We can see that (48) yields

int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

= int

infin

0

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572) 119889120572

le 119862119908 (2119861)int

infin

0

119890minus1198622120572120575119887

1205902lowast119889120572

= 119862119908 (2119861)100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(49)

Thus

100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

le1

119908 (2119861)int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

le119862

119908 (2119861)119908 (2119861)

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast= 119862

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(50)

On the basis of (50) we now estimate 11986812 For the above 120591

select 119906 V such that 1119906 + 1V + 1120591 = 1 by virtue of Holderrsquosinequality and 119908 isin 119860

1 we have

11986812

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times 119908 (119911)1119906 1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119908 (119911)1120591

119908 (119911)minus1+1V

119889119911

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

119906

119908 (119911) 119889119911)

1119906

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minusV+1

119889119911)

1V

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)1

|2119861|119908 (2119861)

1119906+1120591119908 (119909)

minus1+1V|2119861|

1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1V

119908 (119909)minus1+1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(51)

Analogous to the estimate of 11986812 we also have 119868

13le

119862120590lowast119872120591119908

119891(119909)

As for 11986814 taking advantage of Holderrsquos inequality and

(50) we get

11986814

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119908 (119911)

1120591119908 (119911)

11205911015840minus1

119889119911

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|(int

2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

times (int2119861

119908 (119911)1minus1205911015840

119889119911)

11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|119872120591119908

119891 (119909)119908 (2119861)1120591

119908 (119909)11205911015840minus1

|2119861|11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1120591

119908 (119909)11205911015840minus1

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(52)

Collecting the estimates of 11986811 11986812 11986813 and 119868

14 it is easy to see

that

1198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (53)

For term 1198682 using the fact that |119887

2119861minus 119887

119861| le 119862119887

lowast(see

[12]) we now get

1198682le100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

119862

|2119861|int2119861

(1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816)

times1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986821+ 119868

22

(54)

By some estimates similar to those used in the estimate for11986812 we conclude that

11986821

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (55)

For 11986822 using the same method as in dealing with 119868

14 we get

11986822

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (56)

Therefore

1198682le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (57)

Analogously

1198683le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (58)

Journal of Function Spaces 7

An argument similar to that used in the estimate for 11986814leads

to

1198684le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (59)

Hence

119868 le 1198681+ 119868

2+ 119868

3+ 119868

4

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909)

(60)

Next we will consider the second term 119868119868 in (41) For any119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 it is easy to get that |119910 minus 119911| ge 2

119896minus1119903119861

and

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

(61)

Thus

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

int2119896+1119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)119861

10038161003816100381610038161003816sdot sdot sdot

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(62)

For simplicity we also consider the case of 119895 = 2

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681198681+ 119868119868

2+ 119868119868

3+ 119868119868

4

(63)

For 1198681198681 similar to the estimate of 119868

1 we have

1198681198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(64)

Noticing thatwe could use similarmethods of the estimates of1198682 1198683 and 119868

4in the estimates of 119868119868

2 119868119868

3 and 119868119868

4 respectively

From this together with the fact that |1198872119895+1119861minus 119887

119861| le 2

119899(119895 +

1)119887lowast(see [12]) it is easy to get

1198681198682+ 119868119868

3+ 119868119868

4le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (65)

Therefore

119868119868 le 119862120590lowast119872120591119908

119891 (119909) (66)

According to the estimates of 119868 and 119868119868 the lemma has beenproved

Now we will establish a lemma which plays an importantrole in the proof of Theorem 10

Lemma 25 Let 0 lt 120572 lt 119899 119908 isin 1198601 and 119887 isin 119861119872119874(119908) then

for all 119903 gt 1 120591 gt 1 and 119909 isin R119899 we have

119872♯

119871(119871

minus1205722

119891) (119909)

le 119862 1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

119908 (119909)minus120572119899

119872120572119903119908

119891 (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(67)

Proof For any given 119909 isin R119899 take a ball 119861 = 119861(1199090 119903119861) which

contains 119909 For 119891 isin 119871119901(R119899) let 119891

1= 119891120594

2119861 119891

2= 119891 minus 119891

1

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

since 119887 isin BMO(R119899) Using the definition of 119860infin we get

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572)

le 119862119908 (2119861) 119890minus1198622120572120575119887

1205902lowast

(48)

for some 120575 gt 0 We can see that (48) yields

int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

= int

infin

0

119908(119911 isin 2119861 100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816gt 120572) 119889120572

le 119862119908 (2119861)int

infin

0

119890minus1198622120572120575119887

1205902lowast119889120572

= 119862119908 (2119861)100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(49)

Thus

100381610038161003816100381610038161003816(1198871205902

)2119861119908

minus (1198871205902

)2119861

100381610038161003816100381610038161003816

le1

119908 (2119861)int2119861

100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119861

10038161003816100381610038161003816119908 (119911) 119889119911

le119862

119908 (2119861)119908 (2119861)

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast= 119862

100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

(50)

On the basis of (50) we now estimate 11986812 For the above 120591

select 119906 V such that 1119906 + 1V + 1120591 = 1 by virtue of Holderrsquosinequality and 119908 isin 119860

1 we have

11986812

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

times 119908 (119911)1119906 1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119908 (119911)1120591

119908 (119911)minus1+1V

119889119911

le 119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

1

|2119861|(int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

119906

119908 (119911) 119889119911)

1119906

times (int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

(int2119861

119908 (119911)minusV+1

119889119911)

1V

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)1

|2119861|119908 (2119861)

1119906+1120591119908 (119909)

minus1+1V|2119861|

1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1minus1V

119908 (119909)minus1+1V

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(51)

Analogous to the estimate of 11986812 we also have 119868

13le

119862120590lowast119872120591119908

119891(119909)

As for 11986814 taking advantage of Holderrsquos inequality and

(50) we get

11986814

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|int2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119908 (119911)

1120591119908 (119911)

11205911015840minus1

119889119911

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|(int

2119861

1003816100381610038161003816119891 (119911)1003816100381610038161003816

120591

119908 (119911) 119889119911)

1120591

times (int2119861

119908 (119911)1minus1205911015840

119889119911)

11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1

|2119861|119872120591119908

119891 (119909)119908 (2119861)1120591

119908 (119909)11205911015840minus1

|2119861|11205911015840

= 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (119908 (2119861)

|2119861|)

1120591

119908 (119909)11205911015840minus1

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(52)

Collecting the estimates of 11986811 11986812 11986813 and 119868

14 it is easy to see

that

1198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (53)

For term 1198682 using the fact that |119887

2119861minus 119887

119861| le 119862119887

lowast(see

[12]) we now get

1198682le100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

119862

|2119861|int2119861

(1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

+100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816)

times1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

=

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

1003816100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119861119908

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+

119862100381710038171003817100381710038171198871205902

10038171003817100381710038171003817lowast

|2119861|int2119861

100381610038161003816100381610038161003816(1198871205901

)2119861119908

minus (1198871205901

)2119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 11986821+ 119868

22

(54)

By some estimates similar to those used in the estimate for11986812 we conclude that

11986821

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (55)

For 11986822 using the same method as in dealing with 119868

14 we get

11986822

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909) (56)

Therefore

1198682le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (57)

Analogously

1198683le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (58)

Journal of Function Spaces 7

An argument similar to that used in the estimate for 11986814leads

to

1198684le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (59)

Hence

119868 le 1198681+ 119868

2+ 119868

3+ 119868

4

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909)

(60)

Next we will consider the second term 119868119868 in (41) For any119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 it is easy to get that |119910 minus 119911| ge 2

119896minus1119903119861

and

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

(61)

Thus

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

int2119896+1119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)119861

10038161003816100381610038161003816sdot sdot sdot

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(62)

For simplicity we also consider the case of 119895 = 2

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681198681+ 119868119868

2+ 119868119868

3+ 119868119868

4

(63)

For 1198681198681 similar to the estimate of 119868

1 we have

1198681198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(64)

Noticing thatwe could use similarmethods of the estimates of1198682 1198683 and 119868

4in the estimates of 119868119868

2 119868119868

3 and 119868119868

4 respectively

From this together with the fact that |1198872119895+1119861minus 119887

119861| le 2

119899(119895 +

1)119887lowast(see [12]) it is easy to get

1198681198682+ 119868119868

3+ 119868119868

4le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (65)

Therefore

119868119868 le 119862120590lowast119872120591119908

119891 (119909) (66)

According to the estimates of 119868 and 119868119868 the lemma has beenproved

Now we will establish a lemma which plays an importantrole in the proof of Theorem 10

Lemma 25 Let 0 lt 120572 lt 119899 119908 isin 1198601 and 119887 isin 119861119872119874(119908) then

for all 119903 gt 1 120591 gt 1 and 119909 isin R119899 we have

119872♯

119871(119871

minus1205722

119891) (119909)

le 119862 1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

119908 (119909)minus120572119899

119872120572119903119908

119891 (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(67)

Proof For any given 119909 isin R119899 take a ball 119861 = 119861(1199090 119903119861) which

contains 119909 For 119891 isin 119871119901(R119899) let 119891

1= 119891120594

2119861 119891

2= 119891 minus 119891

1

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 7

An argument similar to that used in the estimate for 11986814leads

to

1198684le 119862

10038171003817100381710038171198871205901003817100381710038171003817lowast

119872120591119908

119891 (119909) (59)

Hence

119868 le 1198681+ 119868

2+ 119868

3+ 119868

4

le 1198621003817100381710038171003817119887120590

1003817100381710038171003817lowast119872120591119908

119891 (119909)

(60)

Next we will consider the second term 119868119868 in (41) For any119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 it is easy to get that |119910 minus 119911| ge 2

119896minus1119903119861

and

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816le 119862

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

(61)

Thus

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

int2119896+1119861

1003816100381610038161003816(119887 (119911) minus 119887119861)120590

1003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)119861

10038161003816100381610038161003816sdot sdot sdot

100381610038161003816100381610038161003816119887120590119895(119911) minus (119887

120590119895

)119861

100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(62)

For simplicity we also consider the case of 119895 = 2

119868119868 le 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

(100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816)

times (100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

+10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816)1003816100381610038161003816119891 (119911)

1003816100381610038161003816 119889119911

= 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times100381610038161003816100381610038161198871205902(119911) minus (119887

1205902

)2119896+1119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

100381610038161003816100381610038161198871205901(119911) minus (119887

1205901

)2119896+1119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

+ 119862

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

10038161003816100381610038162119896+1119861

1003816100381610038161003816

times int2119896+1119861

10038161003816100381610038161003816(1198871205901

)2119896+1119861minus (119887

1205901

)119861

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198871205902

)2119896+1119861minus (119887

1205902

)119861

10038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

= 1198681198681+ 119868119868

2+ 119868119868

3+ 119868119868

4

(63)

For 1198681198681 similar to the estimate of 119868

1 we have

1198681198681le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

infin

sum

119896=1

119890minus11986222(119896minus1)

2(119896+1)119899

le 11986210038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909)

(64)

Noticing thatwe could use similarmethods of the estimates of1198682 1198683 and 119868

4in the estimates of 119868119868

2 119868119868

3 and 119868119868

4 respectively

From this together with the fact that |1198872119895+1119861minus 119887

119861| le 2

119899(119895 +

1)119887lowast(see [12]) it is easy to get

1198681198682+ 119868119868

3+ 119868119868

4le 119862

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

119891 (119909) (65)

Therefore

119868119868 le 119862120590lowast119872120591119908

119891 (119909) (66)

According to the estimates of 119868 and 119868119868 the lemma has beenproved

Now we will establish a lemma which plays an importantrole in the proof of Theorem 10

Lemma 25 Let 0 lt 120572 lt 119899 119908 isin 1198601 and 119887 isin 119861119872119874(119908) then

for all 119903 gt 1 120591 gt 1 and 119909 isin R119899 we have

119872♯

119871(119871

minus1205722

119891) (119909)

le 119862 1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

119908 (119909)minus120572119899

119872120572119903119908

119891 (119909)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(67)

Proof For any given 119909 isin R119899 take a ball 119861 = 119861(1199090 119903119861) which

contains 119909 For 119891 isin 119871119901(R119899) let 119891

1= 119891120594

2119861 119891

2= 119891 minus 119891

1

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces

Denote the kernel of 119871minus1205722 by 119870120572(119909 119910) = (120582

1 1205822 120582

119898)

where 120582119895isin R119899 119895 = 1 2 119898 Then 119871

minus1205722

119891 can be written

in the following form

119871minus1205722

119891 (119910)

= intR119899

119898

prod

119895=1

(119887119895(119910) minus 119887

119895(119911))119870

120572(119910 119911) 119891 (119911) 119889119911

= intR119899

119898

prod

119895=1

((119887119895(119910) minus 120582

119895) minus (119887

119895(119911) minus 120582

119895))119870

120572(119910 119911) 119891 (119911) 119889119911

=

119898

sum

119894=0

sum

120590isin119862119898

119894

(minus1)119898minus119894

(119887 (119910) minus 120582)120590

times intR119899

(119887 (119911) minus 120582)1205901015840 119870

120572(119910 119911) 119891 (119911) 119889119911

(68)

Now expanding (119887(119911) minus 120582)1205901015840 as

(119887 (119911) minus 120582)1205901015840 = ((119887 (119911) minus 119887 (119910)) + (119887 (119910) minus 120582))

1205901015840 (69)

and combining (68) with (69) it is easy to see that

119890minus119905119861119871(119871

minus1205722

119891) (119910)

= 119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus 120582

119895) 119871

minus1205722119891) (119910)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 120582)

120590119871minus1205722

1205901015840

119891) (119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

1))(119910)

+ (minus1)119898119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus 120582

119895) 119891

2))(119910)

(70)

where 119905119861= 119903

2

119861and 119903

119861is the radius of ball 119861

Take 120582119895

= (119887119895)119861 119895 = 1 2 119898 and denote

119861=

((1198871)119861 (119887

2)119861 (119887

119898)119861) then

1

|119861|int119861

100381610038161003816100381610038161003816119871minus1205722

119891 (119910) minus 119890

minus119905119861119871(119871

minus1205722

119891) (119910)

100381610038161003816100381610038161003816119889119910

le1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119871

minus1205722119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

(119887 (119910) minus 119887119861)120590119871minus1205722

1205901015840

119891 (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119871

minus1205722119891) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

119890minus119905119861119871((119887 minus 119887

119861)120590119871minus1205722

1205901015840

119891) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

+1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

minus 119890minus119905119861119871(119871

minus1205722(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2))(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

=

7

sum

119894=1

119866119894

(71)

We now estimate the above seven terms respectivelyWe take119898 = 2 as an example the estimate for the case 119898 gt 2 is thesame For the first term 119866

1 we split it as follows

1198661=

1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

100381610038161003816100381610038161198871(119910) minus (119887

1)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

100381610038161003816100381610038161198872(119910) minus (119887

2)119861119908

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

+1

|119861|int119861

10038161003816100381610038161003816(1198871)119861119908

minus (1198871)119861

10038161003816100381610038161003816

10038161003816100381610038161003816(1198872)119861119908

minus (1198872)119861

10038161003816100381610038161003816

10038161003816100381610038161003816119871minus1205722

119891 (119910)10038161003816100381610038161003816119889119910

= 11986611+ 119866

12+ 119866

13+ 119866

14

(72)

Choose 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

Noticing that 119908 isin 1198601 by Holderrsquos inequality and the similar

estimate of 1198681 we have

1198661le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909) (73)

For 1198662 take 120591

1 120591

119895 120591 ] gt 1 that satisfy 1120591

1+ sdot sdot sdot +

1120591119895+ 1120591 + 1] = 1 then following Holderrsquos inequality and

the same idea as that of 1198661yields

1198662le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909) (74)

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 9

To estimate 1198663 applying Lemma 15 (Kolmogorovrsquos ineq-

uality) weak (1 119899(119899 minus 120572)) boundedness of 119871minus1205722 (see

Remark 22) and Holderrsquos inequality we have

1198663=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le119862

|119861|1minus120572119899

10038171003817100381710038171003817100381710038171003817100381710038171003817

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119899(119899minus120572)infin

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119910) minus (119887

119895)119861) 119891 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

(75)

We consider the case of 119898 = 2 for example and we split 1198663

as follows

1198663le

119862

|119861|1minus120572119899

int2119861

(10038161003816100381610038161198871 (119910) minus (119887

1)2119861

1003816100381610038161003816 +1003816100381610038161003816(1198871)2119861

minus (1198871)119861

1003816100381610038161003816)

times (10038161003816100381610038161198872 (119910) minus (119887

2)2119861

1003816100381610038161003816

+1003816100381610038161003816(1198872)2119861

minus (1198872)119861

1003816100381610038161003816)1003816100381610038161003816119891 (119910)

1003816100381610038161003816 119889119910

le119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

10038161003816100381610038161198871 (119910) minus (1198871)2119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

10038161003816100381610038161198872 (119910) minus (1198872)2119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

+119862

|119861|1minus120572119899

int2119861

1003816100381610038161003816(1198871)2119861minus (119887

1)119861

1003816100381610038161003816

1003816100381610038161003816(1198872)2119861minus (119887

2)119861

1003816100381610038161003816

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

= 11986631+ 119866

32+ 119866

33+ 119866

34

(76)

Take 1199031 1199032 119903 119902 gt 1 such that 1119903

1+ 1119903

2+ 1119903 + 1119902 = 1

by virtue of Holderrsquos inequality and the same manner as thatused in dealing with 119868

1 1198682 1198683in Lemma 24 we get

11986631+ 119866

32+ 119866

33le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) (77)

For 11986634

11986634

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1

|119861|1minus120572119899

int2119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909)

(78)

Therefore

1198663le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (79)

By Lemma 24 we have

1198664le 119862

1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872119903119908

(119871minus1205722

119891) (119909)

1198665le 119862

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast119872120591119908

(119871minus1205722

1205901015840

119891) (119909)

(80)

Next we consider the term 1198666

1198666=

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

intR119899

119901119905119861

(119910 119911)

times (119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1))(119911) 119889119911

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

int2119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816119901119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

= 11986661+ 119866

62

(81)

For 11986661 since 119910 isin 119861 119911 isin 2119861 and |119901

119905119861

(119910 119911)| le 119862|2119861|minus1 it

follows that

11986661

le119862

|2119861|int2119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119871minus1205722

(

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

1)(119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 (82)

Analogous to the estimate of 1198663 we have

11986661

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (83)

For 11986662 note that 119910 isin 119861 119911 isin 2

119896+1119861 2

119896119861 and |119901

119905119861

(119910 119911)| le

119862(119890minus11986222(119896minus1)

2(119896+1)119899

|2119896+1

119861|) Hence the estimate of11986662runs as

that of 11986661yields that

11986662

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) + 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909) (84)

For the last term 1198667 applying Lemma 23 we have

1198667le

1

|119861|int119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119871minus1205722

minus 119890minus119905119861119871119871minus1205722

)

times (

119898

prod

119895=1

(119887119895minus (119887

119895)119861) 119891

2)(119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910

le1

|119861|int119861

intR1198992119861

10038161003816100381610038161003816120572119905119861

(119910 119911)10038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911 119889119910

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Function Spaces

le 119862

infin

sum

119896=1

int2119896+11198612119896119861

119905119861

10038161003816100381610038161199090 minus 1199111003816100381610038161003816

119899minus120572+2

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861) 119891 (119911)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862

infin

sum

119896=1

2minus2119896

10038161003816100381610038162119896119861

1003816100381610038161003816

1minus120572119899int2119896+1119861

10038161003816100381610038161003816100381610038161003816100381610038161003816

119898

prod

119895=1

(119887119895(119911) minus (119887

119895)119861)

10038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119891 (119911)1003816100381610038161003816 119889119911

(85)

The same manner as that of 1198663gives us that

1198667le 119862

infin

sum

119896=1

2minus2119896

(1003817100381710038171003817100381710038171003817100381710038171003817lowast

119872120572119903119908

119891 (119909) +1003817100381710038171003817100381710038171003817100381710038171003817lowast

1198721205721119891 (119909))

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

(119872120572119903119908

119891 (119909) +1198721205721119891 (119909))

(86)

According to the above estimates we have completed theproof of Lemma 25

3 Proof of Theorems

At first we give the proof of Theorem 10

Proof It follows from Lemmas 13 14 25 and 17ndash21 that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872119871

minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le100381710038171003817100381710038171003817119872♯

119871(119871minus1205722

119891)

100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119872119903119908

(119871minus1205722

119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

times

1003817100381710038171003817100381710038171003817119872120591119908

(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119908minus120572119899

119872120572119903119908

(119891)10038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+1003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721(119891)

1003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171003817119871minus1205722

11989110038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817119871minus1205722

1205901015840

119891

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198721205721199031199081198911003817100381710038171003817119871119902120581119902119901(119908)

+ 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

+

119898minus1

sum

119895=1

sum

120590isin119862119898

119895

119862119895119898

10038171003817100381710038171003817120590

10038171003817100381710038171003817lowast

1003817100381710038171003817100381710038171003817(119871minus1205722

1205901015840

119891)

1003817100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

(87)

Then we can make use of an induction on 120590 sube 1 2 119898

to get that100381710038171003817100381710038171003817119871minus1205722

119891100381710038171003817100381710038171003817119871119902120581119902119901(119908119902119901 119908)

le 1198621003817100381710038171003817100381710038171003817100381710038171003817lowast

10038171003817100381710038171198911003817100381710038171003817119871119901120581(119908)

(88)

This completes the proof of Theorem 10

Now we are in the position of proving Theorem 11 If wetake 120581 = 0 inTheorem 10 we will immediately get our desiredresults

Conflict of Interests

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

Acknowledgments

Theauthorswould like to express their gratitude to the anony-mous referees for reading the paper carefully and giving somevaluable comments The research was partially supportedby the National Nature Science Foundation of China underGrants nos 11171306 and 11071065 and sponsored by theScientific Project of Zhejiang Provincial Science TechnologyDepartment under Grant no 2011C33012 and the ScientificResearch Fund of Zhejiang Provincial EducationDepartmentunder Grant no Z201017584

References

[1] X T Duong and L X Yan ldquoOn commutators of fractionalintegralsrdquo Proceedings of the American Mathematical Societyvol 132 no 12 pp 3549ndash3557 2004

[2] S Chanillo ldquoA note on commutatorsrdquo Indiana UniversityMathematics Journal vol 31 no 1 pp 7ndash16 1982

[3] H Mo and S Lu ldquoBoundedness of multilinear commutatorsof generalized fractional integralsrdquoMathematische Nachrichtenvol 281 no 9 pp 1328ndash1340 2008

[4] BMuckenhoupt and R LWheeden ldquoWeighted boundedmeanoscillation and the Hilbert transformrdquo StudiaMathematica vol54 no 3 pp 221ndash237 1976

[5] H Wang ldquoOn some commutators theorems for fractionalintegral operators on the weightedMorrey spacesrdquo httparxiv-web3librarycornelleduabs10102638v1

[6] Z Wang and Z Si ldquoCommutator theorems for fractionalintegral operators on weighted Morrey spacesrdquo Abstract andApplied Analysis vol 2014 Article ID 413716 8 pages 2014

[7] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[8] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[9] J Garcıa-Cuerva and J L R de Francia Weighted NormInequalities and Related Topics North-Holland AmsterdamThe Netherlands 1985

[10] JMMartell ldquoSharpmaximal functions associatedwith approx-imations of the identity in spaces of homogeneous type andapplicationsrdquo Studia Mathematica vol 161 no 2 pp 113ndash1452004

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 11

[11] R Johnson and C J Neugebauer ldquoChange of variable resultsfor 119860

119901- and reverse Holder 119877119867

119903-classesrdquo Transactions of the

American Mathematical Society vol 328 no 2 pp 639ndash6661991

[12] A Torchinsky Real-Variable Methods in Harmonic AnalysisAcademic Press San Diego Calif USA 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of