变指标弱型函数空间的实变理论及其应用

Both the real-variable theory of function spaces and the boundedness of operators are always one of the cone contents of harmonic analysis, and, due to the special and plentiful structure, the variable function spaces have been widely used in the studying of applied mathematics such as fluid mechanics, image processing and partial differential equations as well as variational theory. This project aims, based on the work of the applicant about the theory of variable Hardy spaces (associated with operators), variable weak Hardy spaces, variable Besov-type and Trieble-Lizorkin-type spaces and their applications to the boundedness of operators, to establish and develop the theory of variable weak type function spaces including the variable weak local Hardy space、 the variable weak Hardy space associated with operator and the variable weak Besov-type and Trieble-Lizorkin-type spaces associated with operators; particularly, by tools in harmonic analysis such as reproducing formulas and vector-valued inequalities, to characterize those spaces via several maximal functions、atoms、molecules and Littlewood-Paley functions, obtain their real and complex interpolation theorems; moreover, to apply them in studying the boundedness of operators including Calderon-Zygmund operator、fractional integral and their commutators.

函数空间实变理论与算子有界性是调和分析研究领域的核心内容之一, 而变指标函数空间因其独特且丰富的结构被广泛地应用于流体力学、图像处理和偏微分方程及变分理论等应用数学领域研究中. 本项目拟在申请人已有关于(与算子相关的)变指标Hardy空间, 变指标弱Hardy空间, 变指标Besov型和Triebel-Lizorkin型空间的实变理论及其算子有界性的工作基础上, 建立和发展变指标弱型空间实变理论, 其中包括变指标弱局部Hardy空间、与算子相关的变指标弱Hardy空间及变指标弱Besov型和Triebel-Lizorkin型空间；特别地将利用再生公式和向量值不等式等调和分析工具，建立这些空间的各种极大函数刻画、原子和分子特征及Littlewood-Paley函数等特征刻画、实和复插值定理, 并将其应用于包括Calderon-Zygmund算子和分数次积分及其交换子在内的算子有界性的研究中.

数学与物理中的许多重要问题均可归结为算子在某些函数空间中的有界性, 而刻画这些算子的有界性离不开相应函数空间的实变理论. 函数空间实变理论是现代数学的一个重要分支, 它在数学的许多领域如调和分析和偏微分方程中都起着非常重要的作用, 而变指标函数空间因其独特且丰富的结构被广泛地应用于流体力学、图像处理和偏微分方程及变分理论等领域..本项目主要引入了与算子相关的变指标弱Hardy空间并建立其实变理论, 包括极大函数、原子和分子在内的等价特征刻画, 并将其应用于研究算子的有界性. 研究了指标满足更一般条件下的变指标Besov型空间, 建立了它们的球平均特征刻画和Peetre极大函数等价刻画; 引入了与算子相关的变指标Besov空间; 建立了包括Peetre极大函数和热半群在内的等价特征刻画; 另外, 当底空间测度满足逆双倍条件和non-collapsing条件, 建立了该变指标Besov空间的框架分解. 这部分结果, 即使当底空间为一个球的时候也是新的, 且当底空间为欧氏空间及算子为Laplace算子时, 该空间与已有的欧氏空间上变指标Besov空间是一致的 另外, 项目还研究了Triebel-Lizorkin-Morrey空间的复插值问题, 当指标满足一定条件时, 证明了有界Lipschitz区域上Triebel-Lizorkin-Morrey空间的复插值结果是所谓的“diamond”空间.