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 [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-