粗糙核奇异积分及其极大算子的有界性研究

奇异积分及其极大算子的有界性一直是调和分析研究的核心内容之一.本项目将在已有的粗糙核奇异积分及其极大算子的有界性理论基础上，进一步推进与发展上述理论.主要研究内容包括：建立当球面核属于Hardy核条件带粗糙径向核的沿多项式曲线、多项式映射以及解析子流形的奇异积分与极大奇异积分算子在Lebesgue空间、Triebel-Lizorkin空间以及混合径向-角空间上的有界性；建立沿多项式曲线和多项式映射的变量核奇异积分及其极大算子在Lebesgue空间与Triebel-Lizorkin空间上的有界性；研究沿各类低维集的粗糙核奇异积分及其极大算子在混合范数Lebesgue空间上的有界性.

The boundedness of singular integrals and related maximal operators is always one of the core contents of harmonic analysis. This project aims to further promote and develop the boundedness of rough singular integrals and related maximal operators on the base of the previous ones. The main contents of this project are as follows: studying the boundedness of the singular integral operators along polynomial curves, polynomial mappings and real-analytic submanifolds as well as related maximal operators on the Lebesgue spaces, Triebel-Lizorkin spaces and mixed radial-angular spaces under the conditions that the spherical kernel belongs to the Hardy spaces and the radial kernels are rough；establishing the boundedness of the singular integral operators with variable kernels along polynomial curves and supported by polynomial mappings as well as related maximal operators on the Lebesgue spaces and Triebel-Lizorkin spaces；studying the boundedness of rough singular integral operators associated to various lower-dimensional sets and.related maximal operators on the mixed norm Lebesgue spaces.