基于Morrey空间的函数空间实变理论及其应用

项目名称： 基于Morrey空间的函数空间实变理论及其应用

项目编号： No.11471042

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 袁文

作者单位： 北京师范大学

项目金额： 65万元

中文摘要： 数学与物理中的许多重要问题均可归结为研究某些算子在函数空间上的有界性,而刻画这些算子的有界性依赖于相应函数空间的实变理论.申请人与合作者已部分发展了欧氏空间上的Besov型和Triebel-Lizorkin型空间实变理论及其应用.本项目拟进一步完善和发展含欧氏空间在内的度量测度空间上基于Morrey空间的各种函数空间如Besov型和Triebel-Lizorkin型空间等的实变理论,其中包括刻画欧氏空间上这些函数空间的点态乘子、发展度量测度空间上变指标Besov型和Triebel-Lizokrin型空间实变理论、发展欧氏空间上向量值Besov-Morrey空间实变理论并建立相关向量值Fourier乘子定理、建立度量测度空间上相关于热核满足高斯型上界估计、H?lder连续性及统计完备性的算子的Besov型和Triebel-Lizorkin型等空间的实变理论并应用于相关黎斯变换和谱乘子的研究.

中文关键词： 度量测度空间；函数空间；Morrey空间；实变理论；谱乘子

英文摘要： Many important problems arising in mathematics and physics can be reduced to consider the boundedness of some operators on function spaces, and the characterization of these boundedness relies on the real variable theory of the related function spaces. The applicant and his collaborators have partly developed a theory of Besov-type and Triebel-Lizorkin-type spaces on Euclidean spaces and found some of its applications. This project is devoted to complement and develop the real variable theory of some function spaces based on Morrey spaces, including Besov-type and Triebel-Lizorkin-type spaces on Euclidean spaces and metric measure spaces, which contains the following main contents: giving an exact description of the class of pointwise multipliers on these spaces based on Morrey spaces on Euclidean spaces; developing Besov-type and Triebel-Lizorkin-type spaces with variable exponents on metric measure spaces; establishing the real variable theory of operator-valued Besov-Morrey spaces on Euclidean spaces and the related operator-valued Fourier multiplier theorem; developing the real variable theory of Besov-type and Triebel-Lizorkin-type spaces on metric measure spaces associted with operators whose heat kernals satisfy certain Gaussian upper bound estimate, Holder continuity, and the stochastic completeness property, and prove the boundedness of some related Riesz transforms and spectral multipliers on these spaces.

英文关键词： Metric Measure Space;Function Space;Morrey Space;Real Variable Theory;Spectral Multiplier