Boundedness Of Certain Oscillatory Integrals Along Curves Or Surfaces On Sobolev Spaces

This paper is designed to study the boundedness properties of certain oscillatory singular integral operator along curves or surfaces from Sobolev spaces to Lebesgue spaces.The first chapter is intended to introduce the background of oscillatory singular integral operator along curves or surfaces, the present situation of the research and our work of this paper.In chapter two, we study the oscillatory hyper-Hilbert transforms and prove their boundedness properties from Sobolev spaces to Lebesgue spaces.In chapter three, we study certain oscillatory singular integral operator along sur-faces. As applications, we obtain some Sobolev boundedness results of rough singular integral operators on the product spaces.