Boundedness Of Hausdorff Operators On Some Function Spaces

This paper mainly studies the boundedness of several high dimensional Hausdorff operators on some important function spaces. These spaces include Lp spaces, Hp spaces, Herz spaces, the Hardy-type Herz spaces and Triebel-Lizorkin-type space, etc. In addition, this paper contains one chapter concerning the boundedness of oscillatory integrals along curves on Soblev spaces.This paper is divided into five chapters:The first chapter is the introduction, which mainly discusses the development of the Hausdorff operators and the main results of the paper. The Hausdorff operator has a long history, in particular with a close relation to the Cecaro operator and the Hardy operators. One dimensional Hausdorff operator has been thoroughly studied, so this paper focuses the attention on the generalizations in high dimension of the operator. Because of the particularity of its form, the operator has two types of high dimensional generalizations and they could not be transferred to each other simply. One type of generalization was proposed by Liflyand et al in,while the other type is new.The second chapter mainly considers the boundedness of several high dimen-sional Hausdorff operators on the Hardy spaces H1and h1.In,Liflyand and Moricz proved the one dimensional Hausdorff operator is bounded on H1under a simple condition. Later in, Lerner and Liflyand proved the boundedness of one type high dimensional Hausdorff operator on H1. In this chapter, the results on hl are new, while the result on H1improves what is in. All the high dimensional results covers the one dimensional cases.The third chapter studies the boundedness of the high dimensional Hausdorff operators on the Herz and Hardy-type Herz spaces. Here, the Hardy-type Herz spaces are provided with the atomic decompositions, which brings convenient to our proofs. These result are new.The fourth chapter mainly deals with the boundedness of the high dimen- sional Hausdorff operators on the Triebel-Lizorkin-type space Fp.qα,τ. The space is closely related to Triebel-Lizorkin spaces and Qα spaces. Its definition appeared in by Yang et al for the first time. In,Tang et al give its equivalent definition, which actually followed the way of the equivalent characterization of Qp.Qα in.Fp.qα,τcould be regarded as a generalization of Qp.Qα.In,the authors proved the boundedness of the weighted Hardy operators on Fp.qα,τ, and that is the inspiration to the research of this chapter.The fifth chapter presents the boundedness of the oscillatory integral oper-ator along curves on the Soblev spaces. It originated in the Hilbert transform along curves, with a long history. In,Zielinski introduced an oscillatory factor to balance the extra singularity of the hype-Hilbert transform, and in a simple case he proved the corresponding boundedness of the oscillatory integral operator on Lp and L2. And the problems of more general high dimensional cases were solved by Chen, Fan. Wang and Zhu in and eventually. A natural consideration is that if one increases the smoothness of the functions, the re-quirement of the parameters may be reduced. Just on that basis, this chapter considers the corresponding boundedness of the oscillatory integral operator on the Soblev spaces.