Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 946435, 5 pages

http://dx.doi.org/10.1155/2013/946435

Research Article

Boundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces

Yali Pan,¹ Changwen Li,¹ and Xinsong Wang²

¹School of Mathematical Sciences, Huaibei Normal University, Huaibei, Anhui 235000, China

²School of Science, Tianjin Chengjian University, Tianjin 300384, China

Received 17 August 2013; Accepted 11 October 2013

Academic Editor: Yoshihiro Sawano

Copyright © 2013 Yali Pan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Oscillatory integral operators play a key role in harmonic analysis. In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on the weighted Morrey spaces . Meanwhile, the corresponding results for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.

1. Introduction and Main Results

Suppose that is the standard Calderón-Zygmund kernel. That is, is homogeneous of degree , and , where . The oscillatory integral operator is defined by

where , where is the space of infinitely differentiable functions on with compact supports, and is a real-analytic function or a real- function satisfying that, for any , there exists , , , such that does not vanish up to infinite order. These operators have arisen in the study of singular integrals supported on lower dimensional varieties and the singular Radon transform. In [1], Pan proved that are uniform in bounded on . Lu et al. [2] proved the weighted boundedness of defined by (1).

Let be a variable Calderón-Zygmund kernel. That means, for a.e. is a standard Calderón-Zygmund kernel and

Define the oscillatory integral operator with variable Calderón-Zygmund kernel by

where , , and satisfy the same assumptions as those in the operator defined by (1).

Lu et al. [2] investigated the and weighted boundedness about this class of oscillatory integral operators.

The classical Morrey space was first introduced by Morrey in [3] to study the local behavior of solutions to second order elliptic partial differential equations. In 2009, Komori and Shirai [4] first defined the weighted Morrey spaces which could be viewed as an extension of weighted Lebesgue spaces. They studied the boundedness of the fractional integral operator, the Hardy-Littlewood maximal operator, and the Calderón-Zygmund singular integral operator on the space. The boundedness results about some operators on these spaces can be see in ([5–17]). Recently, Shi et al. [18] obtained the boundedness of a class of oscillatory integrals with Calderón-Zygmund kernel and polynomial phase on weighted Morrey spaces. Their results are stated as follows.

Let be a real valued polynomial defined on and let satisfy the following hypotheses:

We define

Theorem A (see [18]). Let , , and . If is of type , then, for any real polynomial , there exists a constant such that

The purpose of this paper is to generalize the above results to the case with real- or analytic phase functions. Our main results in this paper are formulated as follows.

Theorem 1. Let , , and a real- function satisfying that, for any , there exists , , , such that does not vanish up to infinite order. Assume that is a standard Calderón-Zygmund kernel and is defined as in (1). Then for any , , and , is bounded on .

Theorem 2. Let , , and a real- function satisfying that, for any , there exists , , , such that does not vanish up to infinite order. Assume that is a variable Calderón-Zygmund kernel and is defined as in (3). Then for any , , and , is bounded on .

2. Notations and Preliminary Lemmas

Let be the ball with the center and radius . Given a ball and , denotes the ball with the same center as whose radius is times that of .

The classical weighted theory was first introduced by Muckenhoupt in [19]. A weight is a locally integrable function on , which takes values in a.e. For a given weight function , we denote the Lebesgue measure of by and the weighted measure of by ; that is, . Given a weight , we say that satisfies the doubling condition if there exists a constant such that, for any ball , we have .

We say with , if there exists a constant , such that

for every ball . When , if there exists , such that

for almost every . We define . A weight function is said to belong to the reverse Hölder class if there exist two constants and such that the following reverse Hölder inequality holds:

for every ball .

It is well known that, if with , then there exists such that .

Lemma 3 (see [20]). Let , , and . Then for any ball and ,

where does not depend on nor on .

Lemma 4 (see [21]). Let with . Then there exists a constant such that

for any measurable subset of a ball .

The weighted Morrey spaces were defined as follows.

Definition 5 (see [4]). Let , , and a weight function. Then the weighted Morrey space is defined by

where

and the supremum is taken over all balls in . The space is defined by

Our argument is based heavily on the following results.

Lemma 6 (see [2]). Assume that is defined as in (1). Then for any and , one has

where is independent of , , and and .

Lemma 7 (see [2]). Assume that is defined as in (3). Then for any and , one has

where is independent of , , and . is defined in (2).

Definition 8 (see [4]). The Hardy-Littlewood maximal operator is defined by

Lemma 9 (see [4]). If , , and then the Hardy-Littlewood maximal operator is bounded on .

Lemma 10 (see [22]). Denote by the spaces of spherical harmonic functions of degree . Then

(a), and for any ;(b)for any , there exists an orthogonal system of such that , , , and is the Beltrami-Laplace operator on .

In the following the letter will denote a constant which may vary at each occurrence.

3. Proof of Theorems

Proof of Theorem 1. It is sufficient to prove that there exists a constant such that

Fix a ball and decompose , with . Then we have

Using Lemmas 3 and 6, we get

We now estimate . We can write

Now by an argument similar to the proof of Lemma  6 in [2], we choose such that , when , and when . Let and which is large enough and will be determined later. Write

where

Then

Let us first estimate . To do so, using Taylor’s expansion and the compactness of , we write

for, where is a polynomial with deg and with independent of and . Define

Therefore

On , by the properties of and , we have

So we have

By Theorem A and Lemma 9, we have

Now, let us turn to estimate . We consider the following two cases.

Case 1 (). Similar to that estimate of in Lemma 6 in [2], we have

By Lemma 9 we have

Case 2 (). We choose such that

Let

Then

For , by its definition, we can get

The inequality (36) also can be seen in [2]; we omit the details here.

By Lemma 9, we have

Therefore

This finishes the proof of Theorem 1.

Proof of Theorem 2. It is sufficient to prove that there exists a constant such that

Fix a ball and decompose , with . Then we have

Using Lemmas 3 and 7, we get

We now estimate .

For each and , we get

where . Then for a.e. ,

where for any . By Lemma 10, we have that, for any ,

By Lemma 10 again, we can verify that, for any , , and a.e. , if , then

Therefore, from (43), (45), and the Lebesgue dominated convergence theorem, it follows that

We write

It is easy to see that is the oscillatory integral operator defined by (1). By Theorem 1 we have that is bounded on weighted Morrey spaces. Therefore, by (44) and the above discussion we have

This finishes the proof of Theorem 2.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11001001), Natural Science Foundation from the Education Department of Anhui Province (nos. KJ2012B166, KJ2013A235).

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