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Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space

Journal of Inequalities and Applications20112011:479576

Received: 30 June 2010

Accepted: 20 January 2011

Published: 15 February 2011


Let , the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by , where with . The authors prove that the operator is bounded from Sobolev space to space for , and from Hardy-Sobolev space to space for . As corollaries of the result, they also prove the boundedness of the Littlewood-Paley type operators and which relate to the Lusin area integral and the Littlewood-Paley function.


Sobolev SpaceSingular IntegralSingular Integral OperatorAtomic DecompositionHarmonic Polynomial

1. Introduction

Let be the -dimensional Euclidean space and be the unit sphere in equipped with the normalized Lebesgue measure . For , let .

Before stating our theorems, we first introduce some definitions about the variable kernel . A function defined on is said to be in , , if satisfies the following two conditions:

(1) , for any and any ;

(2) .

In 1955, Calderón and Zygmund [1] investigated the boundedness of the singular integrals with variable kernel. They found that these operators connect closely with the problem about the second-order linear elliptic equations with variable coefficients. In 2002, Tang and Yang [2] gave boundedness of the singular integrals with variable kernels associated to surfaces of the form , where for any . That is, they considered the variable Calderón-Zygmund singular integral operator defined by
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 is considered. It is defined by

If , we put . Historically, the higher dimension Marcinkiewicz integral operator with convolution kernel, that is , was first defined and studied by Stein [3] in 1958. See also [46] for some further works on with convolution kernel. Recently, Xue and Yabuta [7] studied the boundedness of the operator with variable kernel.

Theorem 1.1 (see [7]).

Suppose that is positively homogeneous in of degree 0, and satisfies (1.4) and

(2′) , for some . Let be a positive and monotonic (or negative and monotonic) function on and let it satisfy the following conditions:

(i) for some ;

(ii) is monotonic on .

Then there is a constant C such that , where constant is independent of .

Since the condition (2) implies (2′), so the boundedness of holds if with .

Our aim of this paper is to study the hypersingular Marcinkiewicz integral along surfaces with variable kernel , and with index , on the homogeneous Sobolev space for and the homogeneous Hardy-Sobolev space for some . Let be as above, we then define the operators by

Our main results are as follows.

Theorem 1.2.

Suppose that , satisfies (1.4) and with . Let be a positive and increasing function on and let it satisfy the following conditions:

(i) ;

(ii) on .

Then there is a constant C such that , where constant is independent of .

Theorem 1.3.

Suppose , and that , with , and satisfies (1.4). Let be a positive and increasing function on and let it satisfy the following conditions:

(i) ;

(ii) , .

Then, for , there is a constant C such that , where constant is independent of any .

Furthermore, our result can be extended to the Littlewood-Paley type operators and with variable kernels and index , which relate to the Lusin area integral and the Littlewood-Paley function, respectively. Let be as above, we then define the operators and for , respectively by

with , where . As an application of Theorem 1.2, we have the following conclusion.

Theorem 1.4.

Under the assumption of Theorem 1.2, then Theorem 1.2 still holds for and .

By Theorems 1.2 and 1.3 and applying the interpolation theorem of sublinear operator, we obtain the boundedness of .

Corollary 1.5.

Suppose , and that , , and satisfies (1.4). Let be given as in Theorem 1.3. Then, for , there exists an absolute positive constant such that

for all .

Remark 1.6.

It is obvious that the conclusions of Theorem 1.2 are the substantial improvements and extensions of Stein's results in [3] about the Marcinkiewicz integral with convolution kernel, and of Ding's results in [8] about the Marcinkiewicz integral with variable kernels.

Remark 1.7.

Recently, the authors in [9] proved the boundedness of hypersingular Marcinkiewicz integral with variable kernels on homogeneous Sobolev space for and without any smoothness on . So Corollary 1.5 extended the results in [9, Theorem  5].

Throughout this paper, the letter always 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 satisfying and , and if . Let . Define the function by , such that .

For , , and , the homogeneous Triebel-Lizorkin space is the set of all distributions satisfying
For , the homogeneous Sobolev spaces is defined by , namely . From [10] we know that for any
and if is a nonnegative integer, then for any

For , we define the homogeneous Hardy-Sobolev space by . It is well known that for , 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 and has the form where is a solid spherical harmonic polynomial of degree . Then the Fourier transform of has the form , where

and , is the Bessel function.

Lemma 2.2 (see [12]).

For , and , there exists such that for any and ,

Lemma 2.3.

Let , , is a function on and let it satisfy the conditions (i) and (ii) in Theorem 1.2.

Denote , if

Then there exists a constant independent of , such that for every integer .


Let , then we have

So it suffices to show .

Decompose this integral into two parts .

For , by using Lemma 2.2 and , we can get
For the other part , applying Stirling's formula, we have
Also in [13], the authors proved the following inequality
So by (2.9) and (2.10), , and noting that , we have

So far we can deduce the desired conclusion of Lemma 2.3.

Proof of Theorem 1.2.

The basic idea of proof can go back to [14], for recently papers, one see [8, 15]. By the same argument as in [1], let denote the complete system of normalized surface spherical harmonics. See [14] for instance, we can decompose as following:
then we get
Then, applying Hölder inequality twice, we have for any that
By [14, page 230, equation (4.4)], we can observe that the series in the first parenthesis on the right-hand side of the inequality above, for each fixed, is equal to , where is the Sobolev space on with for any . So if we take sufficiently close to 1, then by the Sobolev imbedding theorem , we have

with .

By Fourier transform and (2.16), we get
For , we have
For the integral on the right hand side of the above inequality, by changing of variable, we can get
So we have
Put and , we can deduce from Lemma 2.1 that
Hence, we have

By [14], we know that .

So we can get
Set , and note that , we can deduce that
Noting that is increasing, by using the second mean-value theorem, we get, for some ,
From (2.26), it follows that

Thus using Lemma 2.3, we can deduce the desired conclusion of Theorem 1.2.

Proof of Theorem 1.4.

First, we know that . On the other hand,

Thus, using Theorem 1.2, we can finish Theorem 1.4.

3. The Bounedness on Hardy-Sobolev Spaces

In order to prove the boundedness for operator on Hardy-Sobolev spaces and prove Theorem 1.3, we first introduce a new kind of atomic decomposition for Hardy-Sobolev space as following which will be used next.

Definition 3.1 (see [16]).

For , the function is called a atom if it satisfies the following three conditions:

(1) with a ball ;

(2) ;

(3) , for any polynomial of degree .

By [16], we have that every can be written as a sum of atoms , that is,

Proof of Theorem 1.3.

Similar to the argument of Lemma  3.3 in [17] and using above atomic decomposition, it suffices to show that

with the constant independent of any atom .

Assume . We first note that
For , using Theorem 1.2, it is not difficult to deduce that
For , we first consider the case , according to [15, Lemma  5.5], for and atom with support , one has
Using Minkowski inequality and Hölder inequality for integrals, and (3.5), we can get
For the integral on the right hand side of the above inequality, by changing of variable and noting that , , we can get
By (3.7), we can get
Thus by (3.5) and the condition ,

As for , similar to the argument of , we can easily get . So far the proof of Theorem 1.3 has been finished.



This project supported by the National Natural Science Foundation of China under Grant no. 10747141, Zhejiang Provincial National Natural Science Foundation of China under Grant no. Y604056, and Science Foundation of Shaoguan University under Grant no. 200915001.

Authors’ Affiliations

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


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© W. Ruiying and L. Yin. 2011

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