Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space
© W. Ruiying and L. Yin. 2011
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.
If , we put . Historically, the higher dimension Marcinkiewicz integral operator with convolution kernel, that is , was first defined and studied by Stein  in 1958. See also [4–6] for some further works on with convolution kernel. Recently, Xue and Yabuta  studied the boundedness of the operator with variable kernel.
Theorem 1.1 (see ).
Our main results are as follows.
It is obvious that the conclusions of Theorem 1.2 are the substantial improvements and extensions of Stein's results in  about the Marcinkiewicz integral with convolution kernel, and of Ding's results in  about the Marcinkiewicz integral with variable kernels.
Recently, the authors in  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].
2. The Bounedness on Sobolev Spaces
Before giving the definition of the Sobolev space, let us first recall the Triebel-Lizorkin space.
For , we define the homogeneous Hardy-Sobolev space by . It is well known that for , one can refer  for the details.
Next, let us give the main lemmas we will use in proving theorems.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
So far we can deduce the desired conclusion of Lemma 2.3.
Proof of Theorem 1.2.
By , 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 ).
Proof of Theorem 1.3.
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.
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