research article boundedness of oscillatory hyper-hilbert
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Research ArticleBoundedness of Oscillatory Hyper-HilbertTransform along Curves on Sobolev Spaces
Jun Li1 and Guilian Gao2
1 Department of Mathematics Zhejiang University Hangzhou 310027 China2 School of Science Hangzhou Dianzi University Hangzhou 310018 China
Correspondence should be addressed to Guilian Gao gaoguilian305163com
Received 1 March 2014 Accepted 6 May 2014 Published 19 May 2014
Academic Editor Yongsheng S Han
Copyright copy 2014 J Li and G Gao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The oscillatory hyper-Hilbert transform along curves is of the following form119867119899120572120573
119891(119909) = int1
0
119891(119909minusΓ(119905))119890119894119905minus120573
119905minus1minus120572d119905 where 120572 ge 0
120573 ge 0 and Γ(119905) = (1199051199011 1199051199012 119905
119901119899 ) The study on this operator is motivated by the hyper-Hilbert transform and the strongly
singular integrals The 119871119901 bounds for119867
119899120572120573have been given by Chen et al (2008 and 2010) In this paper for some 120572 120573 and 119901 the
boundedness of119867119899120572120573
on Sobolev spaces 119871119901119904(R119899) and the boundedness of this operator from 119871
2
119904(R119899) to 119871
2
(R119899) are obtained
1 Introduction
In the paper we mainly discuss singular integrals in thefollowing form
119867119899120572120573
119891 (119909) = int
1
0
119891 (119909 minus Γ (119905)) 119890119894119905minus120573
119905minus1minus120572d119905 (1)
where 120572 ge 0 120573 ge 0 and Γ(119905) = (1199051199011 1199051199012 119905
119901119899) denotes a
curve in the n-dimensional spacesOperators of this kind originate from the significant
Hilbert transform
119867119891 (119909) = 119901V intR
119891 (119909 minus 119905)
119905d119905 (2)
In [1] Calderon and Zygmund brought in the rotationmethod shifting the study of the homogeneous singularintegral operators to that of directional Hilbert transforms
119879Ω(119891) (119909) = 119901V int
R119899119891 (119909 minus 119910)
Ω (1199101003816100381610038161003816119910
1003816100381610038161003816)
10038161003816100381610038161199101003816100381610038161003816
119899d119910
=1
2int119878119899minus1
Ω(1199101015840
)1198671199101015840 (119891) (119909) d120590 (119910
1015840
)
(3)
whereΩ is odd and the directional Hilbert transform is
1198671199101015840 (119891) (119909) = 119901V int
R
119891 (119909 minus 1199051199101015840
)d119905119905 (4)
In order to generalize the rotation method Fabes andRiviere [2] introduced the Hilbert transform along curves
119867Γ119891 (119909) = 119901V int
+infin
minusinfin
119891 (119909 minus Γ (119905))d119905119905 (5)
Afterwards the research of119867Γ119891(119909) attractedmany schol-
ars among which Wainger and his fellows contributed to itquite remarkably
Another development derived from Hilbert transform ishypersingular Hilbert transforms
119867120572119891 (119909) = 119901V int
1
minus1
119891 (119909 minus 119905)d119905119905|119905|120572 0 lt 120572 lt 1 (6)
As such operator has more singularity 119891 is required to havesome smoothness It can be proved that 119867
120572is bounded from
119871119901
120572(R119899) to 119871
119901
(R119899) where 1 lt 119901 lt infinA natural question is how to balance the more singularity
due to |119905|120572 without extra smoothness of 119891 Since Hilbert
transform is essentially ldquooscillatoryrdquo we can bring in an oscil-latory factor 119890
119894119905minus120573
in 119867120572 So is the oscillatory hypersingular
integral along curves in the following form
119867119899120572120573
119891 (119909) = int
1
minus1
119891 (119909 minus Γ (119905)) 119890119894|119905|minus120573 d119905
119905|119905|120572 (7)
Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 489068 5 pageshttpdxdoiorg1011552014489068
2 Journal of Function Spaces
where 120572 ge 0 120573 ge 0 and Γ(119905) = (1199051199011 1199051199012 119905
119901119899) denotes a
curve in the n-dimensional spacesIn this direction the thesis of Zielinski [3]was pioneering
In the case 119899 = 2 Γ(119905) = (119905 1199052
) he proved100381710038171003817100381710038171198672120572120573
119891100381710038171003817100381710038171198712(R2)
⪯1003817100381710038171003817119891
10038171003817100381710038171198712(R2)lArrrArr 120573 ge 3120572 (8)
Later on Chandarana [4] generalized the result of Zielin-ski into more common curves showing the correspondingboundedness on 119871
2
(R3) and 119871119901
(R3) However as the com-plexity of his method with the dimension increases he didnot reach a general result
After yearsrsquo exploration the authors in [5] solved thequestion completely
TheoremC (see [5]) Let 120579 = (1205791 1205792 120579
119899) isin R119899 and Γ
120579(119905) =
(1205791|119905|1199011 1205792|119905|1199012 120579
119899|119905|119901119899) Define 119867
119899120572120573as
119867119899120572120573
119891 (119909) = int
1
minus1
119891 (119909 minus Γ120579(119905)) 119890119894|119905|minus120573
|119905|minus1minus120572d119905 120573 gt 120572 (9)
If 1199011 1199012 119901
119899 120572 120573 are all positive
(1) 119867119899120572120573
119891119871119901
(R119899)⪯ 119891
119871119901
(R119899) as long as 120573 gt (119899 +
1)120572 and 2120573(2120573 minus (119899 + 1)120572) lt 119901 lt 2120573(119899 + 1)120572(2) 119867
1198991205721205731198911198712
(R119899)⪯ 119891
1198712
(R119899) 119894119891 120573 = (119899 + 1)120572
Further on the authors [6] proved that if 119901119894are mutually
different then10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712(R119899)
⪯1003817100381710038171003817119891
10038171003817100381710038171198712(R119899)lArrrArr 120573 ge (119899 + 1) 120572 (10)
In [5] it is showed that we only need to consider the partof 119905 ge 0 and Γ
120579(119905) could be reduced to Γ(119905) = (119905
1199011 1199051199012 119905
119901119899)
That is the operator which is given at the very start
119867119899120572120573
119891 (119909) = int
1
0
119891 (119909 minus Γ (119905)) 119890119894119905minus120573
119905minus1minus120572d119905 (11)
and so is what we will discuss in the next section Just underthe bases of [5 6] we probe into the boundedness of 119867
119899120572120573
on Sobolev spaces
2 Preliminary and Main Results
As we know smoothness is a crucial property of functionsand it is common to use high-ordered continuity to describeit Yet an arbitrary function is not always differentiableDue to this Sobolev spaces are introduced to measure thedifferentiability of some more common functions Thesespaces are widely used in both harmonic analysis and PDE
There are several equivalent definitions of such spaces Letus start with the classical definition Firstly we need to recallthe concept of generalized derivatives
Definition 1 Let 119906 isin S1015840 and let 120572 be multiple index Define
⟨120597120572
119906 119891⟩ = (minus1)|120572|
⟨119906 120597|120572|
119891⟩ (12)
If 119906 is a function then 120597120572
119906 the derivative of 119906 in themeaningof distribution is called weak derivative
Definition 2 (see [7]) Let 119896 be a nonnegative integer and 1 lt
119901 lt infin We can define the Sobolev spaces 119871119901119896(R119899) as follows
119871119901
119896(R119899
) = 119891 isin 119871119901
(R119899
) | 120597120572
119891 isin 119871119901
(R119899
) forall120572 |120572| le 119896
(13)
And the norm is given as
119891119871119901
119896
(R119899) = sum
|120572|le119896
1003817100381710038171003817120597120572
1198911003817100381710038171003817119871119901(R119899)
(14)
where 120597(00)
119891 = 119891
It is easy to see that119871119901119896(R119899) is a proper subspace of119871119901(R119899)
The indice 119896 characterizes the smoothness of the functionspaces and we have the following inclusion relations
119871119901
(R119899
) sup 119871119901
1(R119899
) sup 119871119901
2(R119899
) sup 119871119901
3(R119899
) sup sdot sdot sdot (15)
In the above definition 119896 should be an integer Furtheron we can extend the definitions without assuming 119896 to bean integer
Definition 3 (see [7]) Let 119904 be real and 1 lt 119901 lt infin Theinhomogeneous Sobolev spaces 119871
119901
119904(R119899) consisted of all the
elements 119906 of S1015840 which satisfies the following property
((1 +10038161003816100381610038161205851003816100381610038161003816
2
)1199042
) isin 119871119901
(R119899
) (16)
And the corresponding norm is given below
119906 119871119901
119904
(R119899) =
1003817100381710038171003817100381710038171003817((1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
)
1003817100381710038171003817100381710038171003817119871119901(R119899) (17)
For the definition there are some observations
(1) if 119904 = 0 119871119901119904(R119899) = 119871
119901
(R119899)(2) for every 119904 119871119901
119904(R119899) is subset of 119871119901(R119899)
(3) if 119904 = 119896 is a nonnegative integer the two definitionscoincide
Along with inhomogeneous Sobolev spaces we can givethe definition of the homogeneous Sobolev spaces
Definition 4 (see [7]) Let 119904 be a real number and 1 lt 119901 lt infinWe define homogeneous Sobolev spaces 119901
119904(R119899) as follows
119901
119904(R119899
) = 119906 | 119906 isinS1015840 (R119899)
P (
10038161003816100381610038161205851003816100381610038161003816
119904
) isin 119871119901
(R119899
) (18)
and for the distributions in 119901
119904(R119899) we can define
119906119901
119904
(R119899) =1003817100381710038171003817(|sdot|119904
) 1003817100381710038171003817119871119901(R119899)
(19)
What should be noticed is that the elements of homo-geneous Sobolev spaces
119901
119904(R119899) may not belong to 119871
119901
(R119899)Actually these elements are equivalent classes of the temperdistributions Formore details please refer to chapter 6 of [7]
Journal of Function Spaces 3
We also need the following Van der Corput Lemmawhich is themost important lemma to estimate the oscillatingintegrals
Van der Corput Lemma Let120595 and 120601 be smooth real functionsin (119886 119887) and 119896 isin N If |120595(119896)(119905)| ge 1 for all 119905 isin (119886 119887) and oneof the two below conditions are satisfied (1) 119896 = 1 1205951015840(119905) ismonotone in (119886 119887) (2) 119896 ge 2 then
100381610038161003816100381610038161003816100381610038161003816
int
119887
119886
119890119894120582120595(119905)
120601 (119905) d119905100381610038161003816100381610038161003816100381610038161003816
le 119862119896120582minus1119896
(1003816100381610038161003816120601 (119887)
1003816100381610038161003816 + int
119887
119886
100381610038161003816100381610038161206011015840
(119905)10038161003816100381610038161003816d119905)
(20)
The main results of the paper are as follows
Theorem 5 For the operator 119867119899120572120573
in the definition of Γ120579(119905)
1199011 1199012 119901
119899 120572 120573 are all positive If 120573 gt (119899+1)120572 and 2120573(2120573minus
(119899 + 1)120572) lt 119901 lt 2120573(119899 + 1)120572 then10038171003817100381710038171003817119867119899120572120573
11989110038171003817100381710038171003817119871119901
119904
(R119899)⪯
10038171003817100381710038171198911003817100381710038171003817119871119901
119904
(R119899) (21)
Theorem 6 For the operator 119867119899120572120573
in the definition of Γ120579(119905)
1199011 1199012 119901
119899 120572 120573 are all positive If 120572 lt 120573(119873+1(119899 + 1)) and
119873 is the biggest integer not more than 119904 then10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712(R119899)
⪯1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899) (22)
3 Proof of the Main Results
Proof of Theorem 5 To deal with the singularity on thedenominator of the operator 119867
119899120572120573 a dyadic decomposition
is introducedSuppose Φ is a 119862
infin function supported on [12 2] Bynormalization it can be assumed that
+infin
sum
119895=minusinfin
Φ(2119895
119905) equiv 1 (23)
is true for all 119905 gt 0 So we can decomposite119867119899120572120573
as follows
119867119899120572120573
119891 (119909) =
infin
sum
119895=0
int
1
0
Φ(2119895
119905) 119891 (119909 minus Γ (119905)) 119890119894119905minus120573
119905minus1minus120572d119905
=
infin
sum
119895=0
119867119895119891 (119909)
(24)
On account of the support ofΦ we only need to consider thecase where 119895 ge 0
By Minkowskirsquos inequality it is easy to obtain the bound-edness of119867
119895on 1198711
(R119899)
1198671198951198911198711
(R119899) ⪯ 2119895120572
1198911198711
(R119899) (25)
Taking Fourier transform we get the multiple form of119867119895119891
119867119895119891 (120585) = 119898
119895(120585) 119891 (120585) (26)
where
119898119895(120585) = 119898
119895(1205851 1205852 120585
119899)
= int
1
0
Φ(2119895
119905) 119905minus1minus120572
119890119894(119905minus120573
minussum119899
119896=1
120585119896
119905119901
119896 )d119905(27)
In [5] the authors proved
119898119895119871infin
(R119899) ⪯ 2119895(120572minus120573(119899+1))
(28)
Thus by Plancherelrsquos theorem we have
1198671198951198911198712
119904
(R119899) =
1003817100381710038171003817100381710038171003817((1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119867119895119891 (120585))
10038171003817100381710038171003817100381710038171198712(R119899)
=
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119867119895119891(120585)
10038171003817100381710038171003817100381710038171198712(R119899)
=
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119891(120585)119898119895(120585)
10038171003817100381710038171003817100381710038171198712(R119899)
⪯ 2119895(120572minus120573(119899+1))
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119891
10038171003817100381710038171003817100381710038171198712(R119899)
= 2119895(120572minus120573(119899+1))1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899)
(29)
So
10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712119904
(R119899)⪯
infin
sum
119895=0
2119895(120572minus120573(119899+1))1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899) (30)
To make sure 119867119899120572120573
is bounded on 1198712
119904(R119899) (for all 119904) it
is only needed that 120573 gt (119899 + 1)120572 which is the same asthe requirement of the boundedness on 119871
2
(R119899) Roughlyspeaking the operators preserve the smoothness of thefunctions
To get the boundedness on 119871119901
119904(R119899)(119901 gt 1) we will use the
interpolation between (25) and (29) It can be shown that
10038171003817100381710038171003817119867119895
10038171003817100381710038171003817119871119901
2119904(1minus1119901)
(R119899)rarr119871119901
2119904(1minus1119901)
⪯ 2119895(120572minus2120573(1minus1119901)(119899+1))
(31)
As 119904 is arbitrary it suffices to show that120572minus2120573(1minus1119901)(119899+1) lt
0 that is
2120573
2120573 minus (119899 + 1) 120572lt 119901 le 2 (32)
So119867119899120572120573
is bounded on 119871119901
119904(R119899)
By duality argument it is finally proved that if 120573 gt (119899 +
1)120572 then 119867119899120572120573
is bounded on 119871119901
119904(R119899) where 2120573(2120573 minus (119899 +
1)120572) lt 119901 lt 2120573(119899 + 1)120572 and 119904 is arbitrary
Theorem 5 indicates that the operator 119867119899120572120573
can sustainthe ldquosmoothnessrdquo of functions If what we care about is notthe boundedness from Sobolev spaces to Sobolev spaces butthe boundedness from Sobolev spaces to 119871
119901 spaces then thelifting of the smoothness of 119891 can reduce the restriction of 120572120573 which would be explained in the next theorem
4 Journal of Function Spaces
Proof of Theorem 6 Here we will follow the notations andcalculations inTheorem 5 that is
119867119899120572120573
119891 (119909) =
infin
sum
119895=0
119867119895119891 (119909) 119867
119895119891 (120585) = 119898
119895(120585) 119891 (120585)
119898119895(120585) = int
1
0
Φ(2119895
119905) 119905minus1minus120572
119890119894(119905minus120573
minussum119899
119896=1
120585119896
119905119901
119896 )d119905
(33)
Let119873 be the largest integer not exceeding 119904 For Sobolevspaces 1198712
119904(R119899) by Plancherelrsquos theorem when 119904
1gt 1199042
1198712
1199041
(R119899
) sub 1198712
1199042
(R119899
) (34)
and for an element 119891 of 11987121199041
1003817100381710038171003817119891
10038171003817100381710038171198712119904
2
(R119899)lt
100381710038171003817100381711989110038171003817100381710038171198712119904
1
(R119899) (35)
The case 1199041= 119904 1199042= 119873 will be used later
We will make a more accurate estimation of 119898119895 Notice
thatΦ is a119862infin function supported on [12 2] By substitutionof variables 2119895119905 rarr 119905 it is shown that
119898119895(120585) = 2
119895120572
int
infin
0
Φ (119905)
1199051+120572119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d119905 = 2119895120572
119898120572120573
(36)
where we extend the upper limit of the integral into infinityConsidering the support of Φ and 119895 ge 0 this extension willnot make essential difference to the result
In [5] the authors use Van der Corput Lemma and anelementary statement to prove
119898120572120573
119871infin
(R119899) ⪯ 2minus119895120573(119899+1)
(37)
After thoughtful investigation of the proof in [5] it isunearthed that the part Φ(119905)119905
1+120572 will only contribute to thecontrol constant in the inequality above without any effect onthe order of the index
In the subsequent calculation we will substitute the partΦ(119905)119905
1+120572 with notationΨ(119905) AfterwardsΨ(119905) always meansa 119862infin function supported on [12 2] With the process Ψ(119905)
will represent different functions which will not do harm tothe final result That is ifΨ(119905) is a 119862
infin function supported on[12 2] then
10038171003817100381710038171003817100381710038171003817
int
infin
0
Ψ (119905) 119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d11990510038171003817100381710038171003817100381710038171003817119871infin(R119899)
⪯ 2minus119895120573119899+1
(38)
119898120572120573
(120585) =119894
2119895120573int
infin
0
119890minus119894(sum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d1198901198942119895120573
119905minus120573
(39)
using integration by parts
119898120572120573
(120585)
=minus119894
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
+
119899
sum
119896=1
1205851198961199011198962minus119895119901119896
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
(40)
Notice thatΨ indicates different functions in different placesstill they are all 119862infin functions supported on [12 2]
By (38) the absolute value of every integral above can bedominated by 2
minus119895120573(119899+1) Along with Cauchyrsquos inequality wehave
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895120573
(1 +10038161003816100381610038161205851003816100381610038161003816
2
)12
(41)
Repeating integration by parts it is suggested for any119872 that
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895119872120573
(1 + |120585|2
)1198722
(42)
So an estimation to the 1198712 norm of 119867
119895119891 could be made
Recall that119873 represents the largest integer not exceeding 119904
1198671198951198911198712
(R119899) =10038171003817100381710038171003817119867119895119891100381710038171003817100381710038171198712(R119899)
=10038171003817100381710038171003817119898119895119891100381710038171003817100381710038171198712(R119899)
⪯ 2119895120572minus119895119873120573minus119895120573(119899+1)
1003817100381710038171003817100381710038171003817(1 + |120585|
2
)1198732
119891
10038171003817100381710038171003817100381710038171198712(R119899)
= 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119873
(R119899)
le 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899)
(43)
Further on to guarantee 119867119899120572120573
is bounded from 1198712
119904(R119899) to
1198712
(R119899) it is only needed that
120572 minus 119873120573 minus120573
119899 + 1lt 0 (44)
that is 120572 lt 120573(119873 + 1(119899 + 1))When 119904 = 0 119873 = 0 that is 120573 gt (119899 + 1)120572 which is the
result in [5]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by PSF of Zhejiang province(BSH1302046)
References
[1] A P Calderon and A Zygmund ldquoOn singular integralsrdquo Amer-ican Journal of Mathematics vol 78 no 2 pp 289ndash309 1956
[2] E B Fabes and N M Riviere ldquoSingular integrals with mixedhomogeneityrdquo Studia Mathematica vol 27 no 1 pp 19ndash381966
[3] M Zielinski Highly oscillatory singular integrals along curves[PhD dissertation] University of Wisconsin-Madison Madi-son Wis USA 1985
[4] S Chandarana ldquo119871119901-bounds for hypersingular integral opera-tors along curvesrdquo Pacific Journal of Mathematics vol 175 no2 pp 389ndash416 1996
Journal of Function Spaces 5
[5] J ChenD S FanMWang andX R Zhu ldquo119871119901 bounds for oscil-latory hyper-hilbert transform along curvesrdquo Proceedings of theAmerican Mathematical Society vol 136 no 9 pp 3145ndash31532008
[6] J C Chen D S Fan and X R Zhu ldquoSharp 1198712 boundedness
of the oscillatory hyper-Hilbert transform along curvesrdquo ActaMathematica Sinica English Series vol 26 no 4 pp 653ndash6582010
[7] L Grafakos Classical and Modern Fourier Analysis ChinaMachine Press Beijing China 2005
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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2 Journal of Function Spaces
where 120572 ge 0 120573 ge 0 and Γ(119905) = (1199051199011 1199051199012 119905
119901119899) denotes a
curve in the n-dimensional spacesIn this direction the thesis of Zielinski [3]was pioneering
In the case 119899 = 2 Γ(119905) = (119905 1199052
) he proved100381710038171003817100381710038171198672120572120573
119891100381710038171003817100381710038171198712(R2)
⪯1003817100381710038171003817119891
10038171003817100381710038171198712(R2)lArrrArr 120573 ge 3120572 (8)
Later on Chandarana [4] generalized the result of Zielin-ski into more common curves showing the correspondingboundedness on 119871
2
(R3) and 119871119901
(R3) However as the com-plexity of his method with the dimension increases he didnot reach a general result
After yearsrsquo exploration the authors in [5] solved thequestion completely
TheoremC (see [5]) Let 120579 = (1205791 1205792 120579
119899) isin R119899 and Γ
120579(119905) =
(1205791|119905|1199011 1205792|119905|1199012 120579
119899|119905|119901119899) Define 119867
119899120572120573as
119867119899120572120573
119891 (119909) = int
1
minus1
119891 (119909 minus Γ120579(119905)) 119890119894|119905|minus120573
|119905|minus1minus120572d119905 120573 gt 120572 (9)
If 1199011 1199012 119901
119899 120572 120573 are all positive
(1) 119867119899120572120573
119891119871119901
(R119899)⪯ 119891
119871119901
(R119899) as long as 120573 gt (119899 +
1)120572 and 2120573(2120573 minus (119899 + 1)120572) lt 119901 lt 2120573(119899 + 1)120572(2) 119867
1198991205721205731198911198712
(R119899)⪯ 119891
1198712
(R119899) 119894119891 120573 = (119899 + 1)120572
Further on the authors [6] proved that if 119901119894are mutually
different then10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712(R119899)
⪯1003817100381710038171003817119891
10038171003817100381710038171198712(R119899)lArrrArr 120573 ge (119899 + 1) 120572 (10)
In [5] it is showed that we only need to consider the partof 119905 ge 0 and Γ
120579(119905) could be reduced to Γ(119905) = (119905
1199011 1199051199012 119905
119901119899)
That is the operator which is given at the very start
119867119899120572120573
119891 (119909) = int
1
0
119891 (119909 minus Γ (119905)) 119890119894119905minus120573
119905minus1minus120572d119905 (11)
and so is what we will discuss in the next section Just underthe bases of [5 6] we probe into the boundedness of 119867
119899120572120573
on Sobolev spaces
2 Preliminary and Main Results
As we know smoothness is a crucial property of functionsand it is common to use high-ordered continuity to describeit Yet an arbitrary function is not always differentiableDue to this Sobolev spaces are introduced to measure thedifferentiability of some more common functions Thesespaces are widely used in both harmonic analysis and PDE
There are several equivalent definitions of such spaces Letus start with the classical definition Firstly we need to recallthe concept of generalized derivatives
Definition 1 Let 119906 isin S1015840 and let 120572 be multiple index Define
⟨120597120572
119906 119891⟩ = (minus1)|120572|
⟨119906 120597|120572|
119891⟩ (12)
If 119906 is a function then 120597120572
119906 the derivative of 119906 in themeaningof distribution is called weak derivative
Definition 2 (see [7]) Let 119896 be a nonnegative integer and 1 lt
119901 lt infin We can define the Sobolev spaces 119871119901119896(R119899) as follows
119871119901
119896(R119899
) = 119891 isin 119871119901
(R119899
) | 120597120572
119891 isin 119871119901
(R119899
) forall120572 |120572| le 119896
(13)
And the norm is given as
119891119871119901
119896
(R119899) = sum
|120572|le119896
1003817100381710038171003817120597120572
1198911003817100381710038171003817119871119901(R119899)
(14)
where 120597(00)
119891 = 119891
It is easy to see that119871119901119896(R119899) is a proper subspace of119871119901(R119899)
The indice 119896 characterizes the smoothness of the functionspaces and we have the following inclusion relations
119871119901
(R119899
) sup 119871119901
1(R119899
) sup 119871119901
2(R119899
) sup 119871119901
3(R119899
) sup sdot sdot sdot (15)
In the above definition 119896 should be an integer Furtheron we can extend the definitions without assuming 119896 to bean integer
Definition 3 (see [7]) Let 119904 be real and 1 lt 119901 lt infin Theinhomogeneous Sobolev spaces 119871
119901
119904(R119899) consisted of all the
elements 119906 of S1015840 which satisfies the following property
((1 +10038161003816100381610038161205851003816100381610038161003816
2
)1199042
) isin 119871119901
(R119899
) (16)
And the corresponding norm is given below
119906 119871119901
119904
(R119899) =
1003817100381710038171003817100381710038171003817((1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
)
1003817100381710038171003817100381710038171003817119871119901(R119899) (17)
For the definition there are some observations
(1) if 119904 = 0 119871119901119904(R119899) = 119871
119901
(R119899)(2) for every 119904 119871119901
119904(R119899) is subset of 119871119901(R119899)
(3) if 119904 = 119896 is a nonnegative integer the two definitionscoincide
Along with inhomogeneous Sobolev spaces we can givethe definition of the homogeneous Sobolev spaces
Definition 4 (see [7]) Let 119904 be a real number and 1 lt 119901 lt infinWe define homogeneous Sobolev spaces 119901
119904(R119899) as follows
119901
119904(R119899
) = 119906 | 119906 isinS1015840 (R119899)
P (
10038161003816100381610038161205851003816100381610038161003816
119904
) isin 119871119901
(R119899
) (18)
and for the distributions in 119901
119904(R119899) we can define
119906119901
119904
(R119899) =1003817100381710038171003817(|sdot|119904
) 1003817100381710038171003817119871119901(R119899)
(19)
What should be noticed is that the elements of homo-geneous Sobolev spaces
119901
119904(R119899) may not belong to 119871
119901
(R119899)Actually these elements are equivalent classes of the temperdistributions Formore details please refer to chapter 6 of [7]
Journal of Function Spaces 3
We also need the following Van der Corput Lemmawhich is themost important lemma to estimate the oscillatingintegrals
Van der Corput Lemma Let120595 and 120601 be smooth real functionsin (119886 119887) and 119896 isin N If |120595(119896)(119905)| ge 1 for all 119905 isin (119886 119887) and oneof the two below conditions are satisfied (1) 119896 = 1 1205951015840(119905) ismonotone in (119886 119887) (2) 119896 ge 2 then
100381610038161003816100381610038161003816100381610038161003816
int
119887
119886
119890119894120582120595(119905)
120601 (119905) d119905100381610038161003816100381610038161003816100381610038161003816
le 119862119896120582minus1119896
(1003816100381610038161003816120601 (119887)
1003816100381610038161003816 + int
119887
119886
100381610038161003816100381610038161206011015840
(119905)10038161003816100381610038161003816d119905)
(20)
The main results of the paper are as follows
Theorem 5 For the operator 119867119899120572120573
in the definition of Γ120579(119905)
1199011 1199012 119901
119899 120572 120573 are all positive If 120573 gt (119899+1)120572 and 2120573(2120573minus
(119899 + 1)120572) lt 119901 lt 2120573(119899 + 1)120572 then10038171003817100381710038171003817119867119899120572120573
11989110038171003817100381710038171003817119871119901
119904
(R119899)⪯
10038171003817100381710038171198911003817100381710038171003817119871119901
119904
(R119899) (21)
Theorem 6 For the operator 119867119899120572120573
in the definition of Γ120579(119905)
1199011 1199012 119901
119899 120572 120573 are all positive If 120572 lt 120573(119873+1(119899 + 1)) and
119873 is the biggest integer not more than 119904 then10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712(R119899)
⪯1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899) (22)
3 Proof of the Main Results
Proof of Theorem 5 To deal with the singularity on thedenominator of the operator 119867
119899120572120573 a dyadic decomposition
is introducedSuppose Φ is a 119862
infin function supported on [12 2] Bynormalization it can be assumed that
+infin
sum
119895=minusinfin
Φ(2119895
119905) equiv 1 (23)
is true for all 119905 gt 0 So we can decomposite119867119899120572120573
as follows
119867119899120572120573
119891 (119909) =
infin
sum
119895=0
int
1
0
Φ(2119895
119905) 119891 (119909 minus Γ (119905)) 119890119894119905minus120573
119905minus1minus120572d119905
=
infin
sum
119895=0
119867119895119891 (119909)
(24)
On account of the support ofΦ we only need to consider thecase where 119895 ge 0
By Minkowskirsquos inequality it is easy to obtain the bound-edness of119867
119895on 1198711
(R119899)
1198671198951198911198711
(R119899) ⪯ 2119895120572
1198911198711
(R119899) (25)
Taking Fourier transform we get the multiple form of119867119895119891
119867119895119891 (120585) = 119898
119895(120585) 119891 (120585) (26)
where
119898119895(120585) = 119898
119895(1205851 1205852 120585
119899)
= int
1
0
Φ(2119895
119905) 119905minus1minus120572
119890119894(119905minus120573
minussum119899
119896=1
120585119896
119905119901
119896 )d119905(27)
In [5] the authors proved
119898119895119871infin
(R119899) ⪯ 2119895(120572minus120573(119899+1))
(28)
Thus by Plancherelrsquos theorem we have
1198671198951198911198712
119904
(R119899) =
1003817100381710038171003817100381710038171003817((1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119867119895119891 (120585))
10038171003817100381710038171003817100381710038171198712(R119899)
=
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119867119895119891(120585)
10038171003817100381710038171003817100381710038171198712(R119899)
=
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119891(120585)119898119895(120585)
10038171003817100381710038171003817100381710038171198712(R119899)
⪯ 2119895(120572minus120573(119899+1))
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119891
10038171003817100381710038171003817100381710038171198712(R119899)
= 2119895(120572minus120573(119899+1))1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899)
(29)
So
10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712119904
(R119899)⪯
infin
sum
119895=0
2119895(120572minus120573(119899+1))1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899) (30)
To make sure 119867119899120572120573
is bounded on 1198712
119904(R119899) (for all 119904) it
is only needed that 120573 gt (119899 + 1)120572 which is the same asthe requirement of the boundedness on 119871
2
(R119899) Roughlyspeaking the operators preserve the smoothness of thefunctions
To get the boundedness on 119871119901
119904(R119899)(119901 gt 1) we will use the
interpolation between (25) and (29) It can be shown that
10038171003817100381710038171003817119867119895
10038171003817100381710038171003817119871119901
2119904(1minus1119901)
(R119899)rarr119871119901
2119904(1minus1119901)
⪯ 2119895(120572minus2120573(1minus1119901)(119899+1))
(31)
As 119904 is arbitrary it suffices to show that120572minus2120573(1minus1119901)(119899+1) lt
0 that is
2120573
2120573 minus (119899 + 1) 120572lt 119901 le 2 (32)
So119867119899120572120573
is bounded on 119871119901
119904(R119899)
By duality argument it is finally proved that if 120573 gt (119899 +
1)120572 then 119867119899120572120573
is bounded on 119871119901
119904(R119899) where 2120573(2120573 minus (119899 +
1)120572) lt 119901 lt 2120573(119899 + 1)120572 and 119904 is arbitrary
Theorem 5 indicates that the operator 119867119899120572120573
can sustainthe ldquosmoothnessrdquo of functions If what we care about is notthe boundedness from Sobolev spaces to Sobolev spaces butthe boundedness from Sobolev spaces to 119871
119901 spaces then thelifting of the smoothness of 119891 can reduce the restriction of 120572120573 which would be explained in the next theorem
4 Journal of Function Spaces
Proof of Theorem 6 Here we will follow the notations andcalculations inTheorem 5 that is
119867119899120572120573
119891 (119909) =
infin
sum
119895=0
119867119895119891 (119909) 119867
119895119891 (120585) = 119898
119895(120585) 119891 (120585)
119898119895(120585) = int
1
0
Φ(2119895
119905) 119905minus1minus120572
119890119894(119905minus120573
minussum119899
119896=1
120585119896
119905119901
119896 )d119905
(33)
Let119873 be the largest integer not exceeding 119904 For Sobolevspaces 1198712
119904(R119899) by Plancherelrsquos theorem when 119904
1gt 1199042
1198712
1199041
(R119899
) sub 1198712
1199042
(R119899
) (34)
and for an element 119891 of 11987121199041
1003817100381710038171003817119891
10038171003817100381710038171198712119904
2
(R119899)lt
100381710038171003817100381711989110038171003817100381710038171198712119904
1
(R119899) (35)
The case 1199041= 119904 1199042= 119873 will be used later
We will make a more accurate estimation of 119898119895 Notice
thatΦ is a119862infin function supported on [12 2] By substitutionof variables 2119895119905 rarr 119905 it is shown that
119898119895(120585) = 2
119895120572
int
infin
0
Φ (119905)
1199051+120572119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d119905 = 2119895120572
119898120572120573
(36)
where we extend the upper limit of the integral into infinityConsidering the support of Φ and 119895 ge 0 this extension willnot make essential difference to the result
In [5] the authors use Van der Corput Lemma and anelementary statement to prove
119898120572120573
119871infin
(R119899) ⪯ 2minus119895120573(119899+1)
(37)
After thoughtful investigation of the proof in [5] it isunearthed that the part Φ(119905)119905
1+120572 will only contribute to thecontrol constant in the inequality above without any effect onthe order of the index
In the subsequent calculation we will substitute the partΦ(119905)119905
1+120572 with notationΨ(119905) AfterwardsΨ(119905) always meansa 119862infin function supported on [12 2] With the process Ψ(119905)
will represent different functions which will not do harm tothe final result That is ifΨ(119905) is a 119862
infin function supported on[12 2] then
10038171003817100381710038171003817100381710038171003817
int
infin
0
Ψ (119905) 119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d11990510038171003817100381710038171003817100381710038171003817119871infin(R119899)
⪯ 2minus119895120573119899+1
(38)
119898120572120573
(120585) =119894
2119895120573int
infin
0
119890minus119894(sum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d1198901198942119895120573
119905minus120573
(39)
using integration by parts
119898120572120573
(120585)
=minus119894
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
+
119899
sum
119896=1
1205851198961199011198962minus119895119901119896
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
(40)
Notice thatΨ indicates different functions in different placesstill they are all 119862infin functions supported on [12 2]
By (38) the absolute value of every integral above can bedominated by 2
minus119895120573(119899+1) Along with Cauchyrsquos inequality wehave
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895120573
(1 +10038161003816100381610038161205851003816100381610038161003816
2
)12
(41)
Repeating integration by parts it is suggested for any119872 that
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895119872120573
(1 + |120585|2
)1198722
(42)
So an estimation to the 1198712 norm of 119867
119895119891 could be made
Recall that119873 represents the largest integer not exceeding 119904
1198671198951198911198712
(R119899) =10038171003817100381710038171003817119867119895119891100381710038171003817100381710038171198712(R119899)
=10038171003817100381710038171003817119898119895119891100381710038171003817100381710038171198712(R119899)
⪯ 2119895120572minus119895119873120573minus119895120573(119899+1)
1003817100381710038171003817100381710038171003817(1 + |120585|
2
)1198732
119891
10038171003817100381710038171003817100381710038171198712(R119899)
= 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119873
(R119899)
le 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899)
(43)
Further on to guarantee 119867119899120572120573
is bounded from 1198712
119904(R119899) to
1198712
(R119899) it is only needed that
120572 minus 119873120573 minus120573
119899 + 1lt 0 (44)
that is 120572 lt 120573(119873 + 1(119899 + 1))When 119904 = 0 119873 = 0 that is 120573 gt (119899 + 1)120572 which is the
result in [5]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by PSF of Zhejiang province(BSH1302046)
References
[1] A P Calderon and A Zygmund ldquoOn singular integralsrdquo Amer-ican Journal of Mathematics vol 78 no 2 pp 289ndash309 1956
[2] E B Fabes and N M Riviere ldquoSingular integrals with mixedhomogeneityrdquo Studia Mathematica vol 27 no 1 pp 19ndash381966
[3] M Zielinski Highly oscillatory singular integrals along curves[PhD dissertation] University of Wisconsin-Madison Madi-son Wis USA 1985
[4] S Chandarana ldquo119871119901-bounds for hypersingular integral opera-tors along curvesrdquo Pacific Journal of Mathematics vol 175 no2 pp 389ndash416 1996
Journal of Function Spaces 5
[5] J ChenD S FanMWang andX R Zhu ldquo119871119901 bounds for oscil-latory hyper-hilbert transform along curvesrdquo Proceedings of theAmerican Mathematical Society vol 136 no 9 pp 3145ndash31532008
[6] J C Chen D S Fan and X R Zhu ldquoSharp 1198712 boundedness
of the oscillatory hyper-Hilbert transform along curvesrdquo ActaMathematica Sinica English Series vol 26 no 4 pp 653ndash6582010
[7] L Grafakos Classical and Modern Fourier Analysis ChinaMachine Press Beijing China 2005
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
We also need the following Van der Corput Lemmawhich is themost important lemma to estimate the oscillatingintegrals
Van der Corput Lemma Let120595 and 120601 be smooth real functionsin (119886 119887) and 119896 isin N If |120595(119896)(119905)| ge 1 for all 119905 isin (119886 119887) and oneof the two below conditions are satisfied (1) 119896 = 1 1205951015840(119905) ismonotone in (119886 119887) (2) 119896 ge 2 then
100381610038161003816100381610038161003816100381610038161003816
int
119887
119886
119890119894120582120595(119905)
120601 (119905) d119905100381610038161003816100381610038161003816100381610038161003816
le 119862119896120582minus1119896
(1003816100381610038161003816120601 (119887)
1003816100381610038161003816 + int
119887
119886
100381610038161003816100381610038161206011015840
(119905)10038161003816100381610038161003816d119905)
(20)
The main results of the paper are as follows
Theorem 5 For the operator 119867119899120572120573
in the definition of Γ120579(119905)
1199011 1199012 119901
119899 120572 120573 are all positive If 120573 gt (119899+1)120572 and 2120573(2120573minus
(119899 + 1)120572) lt 119901 lt 2120573(119899 + 1)120572 then10038171003817100381710038171003817119867119899120572120573
11989110038171003817100381710038171003817119871119901
119904
(R119899)⪯
10038171003817100381710038171198911003817100381710038171003817119871119901
119904
(R119899) (21)
Theorem 6 For the operator 119867119899120572120573
in the definition of Γ120579(119905)
1199011 1199012 119901
119899 120572 120573 are all positive If 120572 lt 120573(119873+1(119899 + 1)) and
119873 is the biggest integer not more than 119904 then10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712(R119899)
⪯1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899) (22)
3 Proof of the Main Results
Proof of Theorem 5 To deal with the singularity on thedenominator of the operator 119867
119899120572120573 a dyadic decomposition
is introducedSuppose Φ is a 119862
infin function supported on [12 2] Bynormalization it can be assumed that
+infin
sum
119895=minusinfin
Φ(2119895
119905) equiv 1 (23)
is true for all 119905 gt 0 So we can decomposite119867119899120572120573
as follows
119867119899120572120573
119891 (119909) =
infin
sum
119895=0
int
1
0
Φ(2119895
119905) 119891 (119909 minus Γ (119905)) 119890119894119905minus120573
119905minus1minus120572d119905
=
infin
sum
119895=0
119867119895119891 (119909)
(24)
On account of the support ofΦ we only need to consider thecase where 119895 ge 0
By Minkowskirsquos inequality it is easy to obtain the bound-edness of119867
119895on 1198711
(R119899)
1198671198951198911198711
(R119899) ⪯ 2119895120572
1198911198711
(R119899) (25)
Taking Fourier transform we get the multiple form of119867119895119891
119867119895119891 (120585) = 119898
119895(120585) 119891 (120585) (26)
where
119898119895(120585) = 119898
119895(1205851 1205852 120585
119899)
= int
1
0
Φ(2119895
119905) 119905minus1minus120572
119890119894(119905minus120573
minussum119899
119896=1
120585119896
119905119901
119896 )d119905(27)
In [5] the authors proved
119898119895119871infin
(R119899) ⪯ 2119895(120572minus120573(119899+1))
(28)
Thus by Plancherelrsquos theorem we have
1198671198951198911198712
119904
(R119899) =
1003817100381710038171003817100381710038171003817((1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119867119895119891 (120585))
10038171003817100381710038171003817100381710038171198712(R119899)
=
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119867119895119891(120585)
10038171003817100381710038171003817100381710038171198712(R119899)
=
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119891(120585)119898119895(120585)
10038171003817100381710038171003817100381710038171198712(R119899)
⪯ 2119895(120572minus120573(119899+1))
1003817100381710038171003817100381710038171003817(1 +
10038161003816100381610038161205851003816100381610038161003816
2
)1199042
119891
10038171003817100381710038171003817100381710038171198712(R119899)
= 2119895(120572minus120573(119899+1))1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899)
(29)
So
10038171003817100381710038171003817119867119899120572120573
119891100381710038171003817100381710038171198712119904
(R119899)⪯
infin
sum
119895=0
2119895(120572minus120573(119899+1))1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899) (30)
To make sure 119867119899120572120573
is bounded on 1198712
119904(R119899) (for all 119904) it
is only needed that 120573 gt (119899 + 1)120572 which is the same asthe requirement of the boundedness on 119871
2
(R119899) Roughlyspeaking the operators preserve the smoothness of thefunctions
To get the boundedness on 119871119901
119904(R119899)(119901 gt 1) we will use the
interpolation between (25) and (29) It can be shown that
10038171003817100381710038171003817119867119895
10038171003817100381710038171003817119871119901
2119904(1minus1119901)
(R119899)rarr119871119901
2119904(1minus1119901)
⪯ 2119895(120572minus2120573(1minus1119901)(119899+1))
(31)
As 119904 is arbitrary it suffices to show that120572minus2120573(1minus1119901)(119899+1) lt
0 that is
2120573
2120573 minus (119899 + 1) 120572lt 119901 le 2 (32)
So119867119899120572120573
is bounded on 119871119901
119904(R119899)
By duality argument it is finally proved that if 120573 gt (119899 +
1)120572 then 119867119899120572120573
is bounded on 119871119901
119904(R119899) where 2120573(2120573 minus (119899 +
1)120572) lt 119901 lt 2120573(119899 + 1)120572 and 119904 is arbitrary
Theorem 5 indicates that the operator 119867119899120572120573
can sustainthe ldquosmoothnessrdquo of functions If what we care about is notthe boundedness from Sobolev spaces to Sobolev spaces butthe boundedness from Sobolev spaces to 119871
119901 spaces then thelifting of the smoothness of 119891 can reduce the restriction of 120572120573 which would be explained in the next theorem
4 Journal of Function Spaces
Proof of Theorem 6 Here we will follow the notations andcalculations inTheorem 5 that is
119867119899120572120573
119891 (119909) =
infin
sum
119895=0
119867119895119891 (119909) 119867
119895119891 (120585) = 119898
119895(120585) 119891 (120585)
119898119895(120585) = int
1
0
Φ(2119895
119905) 119905minus1minus120572
119890119894(119905minus120573
minussum119899
119896=1
120585119896
119905119901
119896 )d119905
(33)
Let119873 be the largest integer not exceeding 119904 For Sobolevspaces 1198712
119904(R119899) by Plancherelrsquos theorem when 119904
1gt 1199042
1198712
1199041
(R119899
) sub 1198712
1199042
(R119899
) (34)
and for an element 119891 of 11987121199041
1003817100381710038171003817119891
10038171003817100381710038171198712119904
2
(R119899)lt
100381710038171003817100381711989110038171003817100381710038171198712119904
1
(R119899) (35)
The case 1199041= 119904 1199042= 119873 will be used later
We will make a more accurate estimation of 119898119895 Notice
thatΦ is a119862infin function supported on [12 2] By substitutionof variables 2119895119905 rarr 119905 it is shown that
119898119895(120585) = 2
119895120572
int
infin
0
Φ (119905)
1199051+120572119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d119905 = 2119895120572
119898120572120573
(36)
where we extend the upper limit of the integral into infinityConsidering the support of Φ and 119895 ge 0 this extension willnot make essential difference to the result
In [5] the authors use Van der Corput Lemma and anelementary statement to prove
119898120572120573
119871infin
(R119899) ⪯ 2minus119895120573(119899+1)
(37)
After thoughtful investigation of the proof in [5] it isunearthed that the part Φ(119905)119905
1+120572 will only contribute to thecontrol constant in the inequality above without any effect onthe order of the index
In the subsequent calculation we will substitute the partΦ(119905)119905
1+120572 with notationΨ(119905) AfterwardsΨ(119905) always meansa 119862infin function supported on [12 2] With the process Ψ(119905)
will represent different functions which will not do harm tothe final result That is ifΨ(119905) is a 119862
infin function supported on[12 2] then
10038171003817100381710038171003817100381710038171003817
int
infin
0
Ψ (119905) 119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d11990510038171003817100381710038171003817100381710038171003817119871infin(R119899)
⪯ 2minus119895120573119899+1
(38)
119898120572120573
(120585) =119894
2119895120573int
infin
0
119890minus119894(sum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d1198901198942119895120573
119905minus120573
(39)
using integration by parts
119898120572120573
(120585)
=minus119894
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
+
119899
sum
119896=1
1205851198961199011198962minus119895119901119896
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
(40)
Notice thatΨ indicates different functions in different placesstill they are all 119862infin functions supported on [12 2]
By (38) the absolute value of every integral above can bedominated by 2
minus119895120573(119899+1) Along with Cauchyrsquos inequality wehave
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895120573
(1 +10038161003816100381610038161205851003816100381610038161003816
2
)12
(41)
Repeating integration by parts it is suggested for any119872 that
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895119872120573
(1 + |120585|2
)1198722
(42)
So an estimation to the 1198712 norm of 119867
119895119891 could be made
Recall that119873 represents the largest integer not exceeding 119904
1198671198951198911198712
(R119899) =10038171003817100381710038171003817119867119895119891100381710038171003817100381710038171198712(R119899)
=10038171003817100381710038171003817119898119895119891100381710038171003817100381710038171198712(R119899)
⪯ 2119895120572minus119895119873120573minus119895120573(119899+1)
1003817100381710038171003817100381710038171003817(1 + |120585|
2
)1198732
119891
10038171003817100381710038171003817100381710038171198712(R119899)
= 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119873
(R119899)
le 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899)
(43)
Further on to guarantee 119867119899120572120573
is bounded from 1198712
119904(R119899) to
1198712
(R119899) it is only needed that
120572 minus 119873120573 minus120573
119899 + 1lt 0 (44)
that is 120572 lt 120573(119873 + 1(119899 + 1))When 119904 = 0 119873 = 0 that is 120573 gt (119899 + 1)120572 which is the
result in [5]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by PSF of Zhejiang province(BSH1302046)
References
[1] A P Calderon and A Zygmund ldquoOn singular integralsrdquo Amer-ican Journal of Mathematics vol 78 no 2 pp 289ndash309 1956
[2] E B Fabes and N M Riviere ldquoSingular integrals with mixedhomogeneityrdquo Studia Mathematica vol 27 no 1 pp 19ndash381966
[3] M Zielinski Highly oscillatory singular integrals along curves[PhD dissertation] University of Wisconsin-Madison Madi-son Wis USA 1985
[4] S Chandarana ldquo119871119901-bounds for hypersingular integral opera-tors along curvesrdquo Pacific Journal of Mathematics vol 175 no2 pp 389ndash416 1996
Journal of Function Spaces 5
[5] J ChenD S FanMWang andX R Zhu ldquo119871119901 bounds for oscil-latory hyper-hilbert transform along curvesrdquo Proceedings of theAmerican Mathematical Society vol 136 no 9 pp 3145ndash31532008
[6] J C Chen D S Fan and X R Zhu ldquoSharp 1198712 boundedness
of the oscillatory hyper-Hilbert transform along curvesrdquo ActaMathematica Sinica English Series vol 26 no 4 pp 653ndash6582010
[7] L Grafakos Classical and Modern Fourier Analysis ChinaMachine Press Beijing China 2005
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
Proof of Theorem 6 Here we will follow the notations andcalculations inTheorem 5 that is
119867119899120572120573
119891 (119909) =
infin
sum
119895=0
119867119895119891 (119909) 119867
119895119891 (120585) = 119898
119895(120585) 119891 (120585)
119898119895(120585) = int
1
0
Φ(2119895
119905) 119905minus1minus120572
119890119894(119905minus120573
minussum119899
119896=1
120585119896
119905119901
119896 )d119905
(33)
Let119873 be the largest integer not exceeding 119904 For Sobolevspaces 1198712
119904(R119899) by Plancherelrsquos theorem when 119904
1gt 1199042
1198712
1199041
(R119899
) sub 1198712
1199042
(R119899
) (34)
and for an element 119891 of 11987121199041
1003817100381710038171003817119891
10038171003817100381710038171198712119904
2
(R119899)lt
100381710038171003817100381711989110038171003817100381710038171198712119904
1
(R119899) (35)
The case 1199041= 119904 1199042= 119873 will be used later
We will make a more accurate estimation of 119898119895 Notice
thatΦ is a119862infin function supported on [12 2] By substitutionof variables 2119895119905 rarr 119905 it is shown that
119898119895(120585) = 2
119895120572
int
infin
0
Φ (119905)
1199051+120572119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d119905 = 2119895120572
119898120572120573
(36)
where we extend the upper limit of the integral into infinityConsidering the support of Φ and 119895 ge 0 this extension willnot make essential difference to the result
In [5] the authors use Van der Corput Lemma and anelementary statement to prove
119898120572120573
119871infin
(R119899) ⪯ 2minus119895120573(119899+1)
(37)
After thoughtful investigation of the proof in [5] it isunearthed that the part Φ(119905)119905
1+120572 will only contribute to thecontrol constant in the inequality above without any effect onthe order of the index
In the subsequent calculation we will substitute the partΦ(119905)119905
1+120572 with notationΨ(119905) AfterwardsΨ(119905) always meansa 119862infin function supported on [12 2] With the process Ψ(119905)
will represent different functions which will not do harm tothe final result That is ifΨ(119905) is a 119862
infin function supported on[12 2] then
10038171003817100381710038171003817100381710038171003817
int
infin
0
Ψ (119905) 119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )d11990510038171003817100381710038171003817100381710038171003817119871infin(R119899)
⪯ 2minus119895120573119899+1
(38)
119898120572120573
(120585) =119894
2119895120573int
infin
0
119890minus119894(sum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d1198901198942119895120573
119905minus120573
(39)
using integration by parts
119898120572120573
(120585)
=minus119894
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
+
119899
sum
119896=1
1205851198961199011198962minus119895119901119896
2119895120573int
infin
0
119890119894(2119895120573
119905minus120573
minussum119899
119896=1
120585119896
2minus119895119901
119896 119905119901
119896 )
Ψ (119905) d119905
(40)
Notice thatΨ indicates different functions in different placesstill they are all 119862infin functions supported on [12 2]
By (38) the absolute value of every integral above can bedominated by 2
minus119895120573(119899+1) Along with Cauchyrsquos inequality wehave
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895120573
(1 +10038161003816100381610038161205851003816100381610038161003816
2
)12
(41)
Repeating integration by parts it is suggested for any119872 that
10038161003816100381610038161003816119898120572120573
(120585)10038161003816100381610038161003816⪯ 2minus119895120573(119899+1)
2minus119895119872120573
(1 + |120585|2
)1198722
(42)
So an estimation to the 1198712 norm of 119867
119895119891 could be made
Recall that119873 represents the largest integer not exceeding 119904
1198671198951198911198712
(R119899) =10038171003817100381710038171003817119867119895119891100381710038171003817100381710038171198712(R119899)
=10038171003817100381710038171003817119898119895119891100381710038171003817100381710038171198712(R119899)
⪯ 2119895120572minus119895119873120573minus119895120573(119899+1)
1003817100381710038171003817100381710038171003817(1 + |120585|
2
)1198732
119891
10038171003817100381710038171003817100381710038171198712(R119899)
= 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119873
(R119899)
le 2119895120572minus119895119873120573minus119895120573(119899+1)1003817100381710038171003817119891
10038171003817100381710038171198712119904
(R119899)
(43)
Further on to guarantee 119867119899120572120573
is bounded from 1198712
119904(R119899) to
1198712
(R119899) it is only needed that
120572 minus 119873120573 minus120573
119899 + 1lt 0 (44)
that is 120572 lt 120573(119873 + 1(119899 + 1))When 119904 = 0 119873 = 0 that is 120573 gt (119899 + 1)120572 which is the
result in [5]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was supported by PSF of Zhejiang province(BSH1302046)
References
[1] A P Calderon and A Zygmund ldquoOn singular integralsrdquo Amer-ican Journal of Mathematics vol 78 no 2 pp 289ndash309 1956
[2] E B Fabes and N M Riviere ldquoSingular integrals with mixedhomogeneityrdquo Studia Mathematica vol 27 no 1 pp 19ndash381966
[3] M Zielinski Highly oscillatory singular integrals along curves[PhD dissertation] University of Wisconsin-Madison Madi-son Wis USA 1985
[4] S Chandarana ldquo119871119901-bounds for hypersingular integral opera-tors along curvesrdquo Pacific Journal of Mathematics vol 175 no2 pp 389ndash416 1996
Journal of Function Spaces 5
[5] J ChenD S FanMWang andX R Zhu ldquo119871119901 bounds for oscil-latory hyper-hilbert transform along curvesrdquo Proceedings of theAmerican Mathematical Society vol 136 no 9 pp 3145ndash31532008
[6] J C Chen D S Fan and X R Zhu ldquoSharp 1198712 boundedness
of the oscillatory hyper-Hilbert transform along curvesrdquo ActaMathematica Sinica English Series vol 26 no 4 pp 653ndash6582010
[7] L Grafakos Classical and Modern Fourier Analysis ChinaMachine Press Beijing China 2005
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
[5] J ChenD S FanMWang andX R Zhu ldquo119871119901 bounds for oscil-latory hyper-hilbert transform along curvesrdquo Proceedings of theAmerican Mathematical Society vol 136 no 9 pp 3145ndash31532008
[6] J C Chen D S Fan and X R Zhu ldquoSharp 1198712 boundedness
of the oscillatory hyper-Hilbert transform along curvesrdquo ActaMathematica Sinica English Series vol 26 no 4 pp 653ndash6582010
[7] L Grafakos Classical and Modern Fourier Analysis ChinaMachine Press Beijing China 2005
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