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

- Wei Ruiying
^{1}Email author and - Li Yin
^{1}

**2011**:479576

https://doi.org/10.1155/2011/479576

© W. Ruiying and L. Yin. 2011

**Received: **30 June 2010

**Accepted: **20 January 2011

**Published: **15 February 2011

## Abstract

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.

## 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:

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 [4–6] 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:

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 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:

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:

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

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.

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 , 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]).

Lemma 2.2 (see [12]).

Lemma 2.3.

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

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

Proof.

Decompose this integral into two parts .

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

Proof of Theorem 1.2.

By [14], we know that .

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

Proof of Theorem 1.4.

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:

(3) , for any polynomial of degree .

Proof of Theorem 1.3.

with the constant independent of any atom .

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

## Declarations

### Acknowledgments

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

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