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

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:

(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

(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 is considered. It is defined by

(1.2)

where

(1.3)

(1.4)

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:

(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

(1.5)

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

(1.6)

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

(1.7)

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

(2.1)

For , the homogeneous Sobolev spaces is defined by , namely . From [10] we know that for any

(2.2)

and if is a nonnegative integer, then for any

(2.3)

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

(2.4)

and , is the Bessel function.

Lemma 2.2 (see [12]).

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

(2.5)

Lemma 2.3.

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

Denote , if

(2.6)

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

Proof.

Let , then we have

(2.7)

So it suffices to show .

Decompose this integral into two parts .

For , by using Lemma 2.2 and , we can get

(2.8)

For the other part , applying Stirling's formula, we have

(2.9)

Also in [13], the authors proved the following inequality

(2.10)

So by (2.9) and (2.10), , and noting that , we have

(2.11)

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:

(2.12)

Denote

(2.13)

then we get

(2.14)

Then, applying Hölder inequality twice, we have for any that

(2.15)

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

(2.16)

with .

By Fourier transform and (2.16), we get

(2.17)

For , we have

(2.18)

For the integral on the right hand side of the above inequality, by changing of variable, we can get

(2.19)

So we have

(2.20)

Put and , we can deduce from Lemma 2.1 that

(2.21)

where

(2.22)

Hence, we have

(2.23)

By [14], we know that .

So we can get

(2.24)

Set , and note that , we can deduce that

(2.25)

Noting that is increasing, by using the second mean-value theorem, we get, for some ,

(2.26)

From (2.26), it follows that

(2.27)

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,

(2.28)

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,

(3.1)

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

(3.2)

with the constant independent of any atom .

Assume . We first note that

(3.3)

For , using Theorem 1.2, it is not difficult to deduce that

(3.4)

For , we first consider the case , according to [15, Lemma  5.5], for and atom with support , one has

(3.5)

Using Minkowski inequality and Hölder inequality for integrals, and (3.5), we can get

(3.6)

For the integral on the right hand side of the above inequality, by changing of variable and noting that , , we can get

(3.7)

By (3.7), we can get

(3.8)

Thus by (3.5) and the condition ,

(3.9)

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