Morrey Spaces and Related Function SpacesView this Special Issue
Boundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces
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 , Pan proved that are uniform in bounded on . Lu et al.  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.  investigated the and weighted boundedness about this class of oscillatory integral operators.
The classical Morrey space was first introduced by Morrey in  to study the local behavior of solutions to second order elliptic partial differential equations. In 2009, Komori and Shirai  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.  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 ). 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 . 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 ). Let , , and . Then for any ball and , where does not depend on nor on .
Lemma 4 (see ). 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 ). 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.
Definition 8 (see ). The Hardy-Littlewood maximal operator is defined by
Lemma 9 (see ). If , , and then the Hardy-Littlewood maximal operator is bounded on .
Lemma 10 (see ). 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 , 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 , 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 ; 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.
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).
J. Peetre, “On the theory of spaces,” Journal of Functional Analysis, vol. 4, no. 1, pp. 71–87, 1969.View at: Google Scholar
S. Z. Lu, Y. Ding, and D. Y. Yan, Singular Integrals and Related Topics, World Scientific Publishing, River Edge, NJ, USA, 2007.
H. Wang, “The boundedness of some operators with rough kernel on the weighted Morrey spaces,” Acta Mathematica Sinica, Chinese Series, vol. 55, no. 4, pp. 589–600, 2012 (Chinese).View at: Google Scholar
H. Wang, “The boundedness of fractional integral operators with rough kernels on the weighted Morrey spaces,” Acta Mathematica Sinica, Chinese Series, vol. 56, no. 2, pp. 175–186, 2013 (Chinese).View at: Google Scholar
S. G. Shi, Z. W. Fu, and S. Z. Lu, “Boundedness of oscillatory integral operators and their commutators on weighted Morrey spaces,” Scientia Sinica Mathematica, vol. 43, pp. 147–158, 2013 (Chinese).View at: Google Scholar
J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.View at: MathSciNet
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, USA, 1971.