Báo cáo hóa học: " Research Article Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space"

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2011, Article ID 479576, 14 pages

doi:10.1155/2011/479576

Research Article

Hypersingular Marcinkiewicz Integrals along

Surface with Variable Kernels on Sobolev Space

and Hardy-Sobolev Space

Wei Ruiying and Li Yin

School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China

Correspondence should be addressed to Wei Ruiying, weiruiying521@163.com

Received 30 June 2010; Revised 5 December 2010; Accepted 20 January 2011

Academic Editor: Andrei Volodin

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Let α≥0, the authors introduce in this paper a class of the hypersingular

Marcinkiewicz integrals along surface with variable kernels defined by µΦ

Ω,αfx

∞

0||y|≤tΩx, y/|y|n−1fx−Φ|y|y′dy|2dt/t32α1/2,whereΩx, z∈L∞

n×Lq

n−1

with q>max{1,2n−1/n2α}. The authors prove that the operator µΦ

Ω,α is bounded from

Sobolev space Lp

α

nto Lp

nspace for 1 <p≤2, and from Hardy-Sobolev space Hp

α

nto

Lp

nspace for n/nα<p≤1. As corollaries of the result, they also prove the ˙

αRn−L2Rn

boundedness of the Littlewood-Paley type operators µΦ

Ω,α,S and µ∗,Φ

Ω,α,λ which relate to the Lusin

area integral and the Littlewood-Paley g∗

λfunction.

1. Introduction

Let

nn≥2be the n-dimensional Euclidean space and

n−1be the unit sphere in

equipped with the normalized Lebesgue measure dσ dσ·.Forx∈

n\{0},letx′x/|x|.

Before stating our theorems, we first introduce some definitions about the variable

kernel Ωx, z.AfunctionΩx, zdefined on

n×

nis said to be in L∞

n×Lq

n−1,q≥1,

if Ωx, zsatisfies the following two conditions:

1Ωx, λzΩx, z,foranyx, z ∈

nand any λ>0;

2ΩL∞

n×Lq

n−1supr≥0,y∈

n

n−1|Ωrz′y, z′|qdσz′1/q <∞.

In 1955, Calder´

on and Zygmund 1investigated the Lpboundedness of the singular

integrals TΩwith variable kernel. They found that these operators connect closely with the

2 Journal of Inequalities and Applications

problem about the second-order linear elliptic equations with variable coeﬃcients. In 2002,

Tang and Yang 2gave Lpboundedness of the singular integrals with variable kernels

associated to surfaces of the form {xΦ|y|y′},wherey′y/|y|for any y∈

n\{0}n≥

2. That is, they considered the variable Calder ´

on-Zygmund singular integral operator TΦ

defined by

TΦ

Ωfxp·v·

Ωx, y



y



nfx−Φ

y

y′dy. 1.1

On the other hand, as a related vector-valued singular integral with variable kernel,

the Marcinkiewicz singular with rough variable kernel associated with surfaces of the form

{xΦ|y|y′}is considered. It is defined by

µΦ

Ωfx∞

0



FΦ

Ω,tx



2dt

t31/2

,1.2

where

FΦ

Ω,tx|y|≤t

Ωx, y



y



n−1fx−Φ

y

y′dy, 1.3



n−1

Ωx, z′dσz′0.1.4

If Φ|y||y|,weputµΦ

ΩµΩ. Historically, the higher dimension Marcinkiewicz

integral operator µΩwith convolution kernel, that is Ωx, zΩz, was first defined and

studied by Stein 3in 1958. See also 4–6for some further works on µΩwith convolution

kernel. Recently, Xue and Yabuta 7studied the L2boundedness of the operator µΦ

Ωwith

variable kernel.

Theorem 1.1 see 7.Suppose that Ωx, yis positively homogeneous in yof degree 0, and satisfies

1.4and

2′supy∈

n

n−1|Ωy, z′|qdσz′1/q <∞,forsomeq>2n−1/n.LetΦbe a positive

and monotonic (or negative and monotonic) C1function on 0,∞and let it satisfy the

following conditions:

iδ≤|Φt/tΦ′t|≤Mfor some 0<δ≤M<∞;

iiΦ

′tis monotonic on 0,∞.

Then there is a constant C such that µΦ

Ωf2≤Cf2, where constant Cis independent of f.

Since the condition 2implies 2′,sotheL2

nboundedness of µΦ

Ωholds if Ω∈

L∞

n×Lq

n−1with q>2n−1/n.

Our aim of this paper is to study the hypersingular Marcinkiewicz integral µΦ

Ω,α along

surfaces with variable kernel Ω,andwithindexα≥0, on the homogeneous Sobolev space

Journal of Inequalities and Applications 3

α

nfor 1 <p≤2 and the homogeneous Hardy-Sobolev space Hp

α

nfor some n/nα<

p≤1. Let FΦ

Ω,txbe as above, we then define the operators µΦ

Ω,α by

µΦ

Ω,αfx∞

0



FΦ

Ω,tx



2dt

t32α1/2

,α≥0.1.5

Our main results are as follows.

Theorem 1.2. Suppose that α≥0,Ωx, ysatisfies 1.4and Ω∈L∞

n×Lq

n−1with q>

max{1,2n−1/n2α}.LetΦbe a positive and increasing C1function on 0,∞and let it satisfy

the following conditions:

iΦt≃tΦ′t;

ii0≤Φ′t≤Won 0,∞.

Then there is a constant C such that µΦ

Ω,αfL2

n≤CfL2

α

n, where constant Cis independent

of f.

Theorem 1.3. Suppose 0<α<n/2,andthatΩ∈L∞

n×LqSn−1,withq>max{1,2n−

1/n2α}, and satisfies 1.4.LetΦbe a positive and increasing C1function on 0,∞and let it

satisfy the following conditions:

iΦt≃tΦ′t;

ii0<Φ′t≤1,Φ00.

Then, for n/nα<p≤1, there is a constant C such that µΦ

Ω,αfLp

n≤CfHp

α

n,

where constant Cis independent of any f∈Hp

α

n∩S

n.

Furthermore, our result can be extended to the Littlewood-Paley type operators µΦ

Ω,α,S

and µ∗,Φ

Ω,α,λ with variable kernels and index α≥0, which relate to the Lusin area integral and

the Littlewood-Paley g∗

λfunction, respectively. Let FΦ

Ω,txbe as above, we then define the

operators µΦ

Ω,α,S and µ∗,Φ

Ω,α,λ for f∈S

n, respectively by

µΦ

Ω,α,SfxΓx



FΦ

Ω,ty



2dydt

tn32α1/2

µ∗,Φ

Ω,α,λfx⎛

⎝

n1

t

t

x−y

λn



FΦ

Ω,ty



2dydt

tn32α⎞

⎠

1/2

1.6

with λ>1, where Γx{y, t∈

n1

:|x−y|<t}. As an application of Theorem 1.2,we

have the following conclusion.

Theorem 1.4. Under the assumption of Theorem 1.2,thenTheorem 1.2 still holds for µΦ

Ω,α,S and

µ∗,Φ

Ω,α,λ.

By Theorems 1.2 and 1.3 and applying the interpolation theorem of sublinear operator,

we obtain the Lp

α−Lpboundedness of µΦ

Ω,α.

4 Journal of Inequalities and Applications

Corollary 1.5. Suppose 0<α<n/2,andthatΩ∈L∞

n×LqSn−1,q>max{1,2n−1/n

2α}, and satisfies 1.4.LetΦbe given as in Theorem 1.3.Then,for1<p≤2, there exists an absolute

positive constant Csuch that



µΦ

Ω,αf



Lp

n≤C

f

Lp

α

n,1.7

for all f∈Lp

α

n∩S

n.

Remark 1.6. It is obvious that the conclusions of Theorem 1.2 are the substantial improve-

ments and extensions of Stein’s results in 3about the Marcinkiewicz integral µΩwith

convolution kernel, and of Ding’s results in 8about the Marcinkiewicz integral µΩwith

variable kernels.

Remark 1.7. Recently, the authors in 9proved the boundedness of hypersingular

Marcinkiewicz integral with variable kernels on homogeneous Sobolev space Lp

αRnfor

1<p≤2and0<α<1 without any smoothness on Ω.SoCorollary 1.5 extended the results

in 9,Theorem5.

Throughout this paper, the letter Calways remains to denote a positive constant not

necessarily the same at each occurrence.

2. The Bounedness on Sobolev Spaces

Before giving the definition of the Sobolev space, let us first recall the Triebel-Lizorkin space.

Fix a radial function ϕx∈C∞satisfying suppϕ⊆{x:1/2<|x|≤2}and 0 ≤

ϕx≤1, and ϕx>c>0if3/5≤|x|≤5/3. Let ϕjxϕ2jx.Definethefunctionψjxby

Fψjξϕjξ,suchthatFψj∗fξFfξϕjξ.

For 0 <p, q<∞,andα∈

, the homogeneous Triebel-Lizorkin space ˙

Fα,q

pis the set of

all distributions fsatisfying

Fα,q

p

n⎧

⎪

⎨

⎪

⎩

f∈S

′

n:

f

˙

Fα,q

p





k



2−αkψk∗f



q1/q



p

<∞⎫

⎪

⎬

⎪

⎭

.2.1

For p≥1, the homogeneous Sobolev spaces Lp

α

nis defined by Lp

α

n ˙

Fα,2

p

n,

namely fLp

αf˙

Fα,2

p.From10we know that for any f∈L2

α

n



f

L2

α

n∼



n

Ffξ



2|ξ|2αdξ1/2

,2.2

and if αis a nonnegative integer, then for any f∈Lp

α

n



f

Lp

α

n∼



|τ|α

Dτf

Lp

n.2.3

Journal of Inequalities and Applications 5

For 0 <p≤1, we define the homogeneous Hardy-Sobolev space Hp

α

nby Hp

α

n

Fα,2

p

n.ItiswellknownthatHp

n ˙

F0,2

p

nfor 0 <p≤1, one can refer 10for the

details.

Next, let us give the main lemmas we will use in proving theorems.

Lemma 2.1 see 11.Suppose that n≥2and f∈L1

n∩L2

nhas the form fx

f0|x|Pxwhere Pxis a solid spherical harmonic polynomial of degree m. Then the Fourier

transform of fhas the form FfxF0|x|Px,where

F0r2πi−mr−n2m−2/2∞

f0sJn2m−2/22πrssn2m/2ds, 2.4

and r|ξ|,Jmsis the Bessel function.

Lemma 2.2 see 12.For λn−2/2,and−λ≤α≤1,thereexistsC>0such that for any

h≥0and m1,2,...,



h

Jmλt

tλαdt



≤C

mλα.2.5

Lemma 2.3. Let α≥0,λn−2/2,Φis a C1function on 0,∞and let it satisfy the conditions

(i) and (ii) in Theorem 1.2.

Denote gαfx

∞

0|Nεfx|2dε/ε12α1/2,if

FNεfξΦε|ξ|

Jmλt

tλ1dt ·F

fξ.2.6

Then there exists a constant Cindependent of m,suchthatgαfL2≤C/mλ1αfL2

αfor every

integer m∈

,m >α.

Proof. Let η|x||x|

0Jmλt/tλ1dt,thenwehave



gαf



2

n∞

0

Nεfx



2dε

ε12αdx

∞

0

n

ηΦε|ξ|Ffξ



2dξ dε

ε12α

∞

0

n



ηΦβ

|ξ||ξ|Ffξ



|ξ|2αdξ dβ

β12α



n∞

0



ηΦβ

|ξ||ξ|



2dβ

β12α

Ffξ



2|ξ|2αdξ.

2.7

So it suﬃces to show ∞

0ηΦβ/|ξ||ξ|2dβ/β12α≤C/mλ1α2.

Decompose this integral into two parts ∞

0m/2

0∞

m/2:I1I2.

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