齐型空间上混合范数函数空间实变理论及其应用

齐型空间,由于涵盖了调和分析中众多经典底空间,从而具有广泛一般性和根本重要性;其上的函数空间实变理论及其应用一直是调和分析核心研究内容之一.由于混合范数勒贝格空间有着比经典勒贝格空间更加精细的结构,已被广泛应用于数学和物理中的各种分析问题的研究中.本项目拟在申请人已有关于函数空间实变理论及其应用研究工作的基础上,进一步发展一套完整的齐型空间上混合范数Hardy空间实变理论,其中包括各种极大函数、原子或分子以及Littlewood-Paley函数特征刻画,对偶空间等.由此发展一套完整的齐型空间上具有光滑指标的混合范数Besov空间与Triebel-Lizorkin空间的实变理论,包括其小波和原子或分子特征刻画,建立其与经典函数空间的关系,并将其应用于奇异积分等算子的有界性的研究中.这些结果将丰富齐型空间上精细函数空间实变理论,为进一步发展度量测度空间上的分析提供更为精细的工作空间和理论工具.

The spaces of homogeneous type cover many classical underlying spaces in harmonic analysis, thus having wide generality and fundamental importance. The real-variable theory of function spaces on these spaces and their applications has been a core research topic in harmonic analysis. Due to its more refined structure compared to classical Lebesgue spaces, the mixed norm Lebesgue spaces have been widely used in the study of various analytical problems in mathematics and physics. Based on the applicant's existing research on the theory and applications of function spaces in real analysis, this project intends to develop a complete real-variable theory of mixed-norm Hardy spaces on spaces of homogeneous type, including characterizations of various maximal functions, atoms or molecules, and Littlewood-Paley function, as well as dual spaces. This will lead to a complete theory of mixed-norm Besov and Triebel-Lizorkin spaces with smooth indices on spaces of homogeneous type, including characterizations of their wavelet and atom or molecule characterization, establishing their relationship with classical function spaces, and applying them to the study of the boundedness of singular integral operators. These results will enrich the theory of fine function spaces on spaces of homogeneous type and provide more refined working spaces and theoretical tools for further development of analysis on metric measure spaces.