函数空间实变理论与算子有界性研究一直是调和分析的核心内容之一.本项目拟在申请人及其合作者已有关于变指标、Musielak-Orlicz型等函数空间实变理论与算子有界性研究的基础上,通过充分揭示底空间几何性质的分析表达形式,并将其与基础函数空间的精细结构巧妙结合起来,以及运用来自调和分析、泛函分析等相关分析学科的一些基本理论和技巧,以获得建立函数空间实变理论所必需的基本工具;由此发展一套关于齐型空间等各种底空间上的广义混合范数(变指标)函数空间和相关于Musielak-Orlicz-slice空间等基础函数空间的包括各种精细特征刻画在内的函数空间实变理论;应用这些特征进一步建立相应Hardy-型空间及其对偶空间乘积的双线性分解,并由此研究交换子有界性,散度-旋度引理及广义Leibniz-型法则等.这些研究将进一步丰富具有精细结构的函数空间实变理论,并为调和分析、偏微分方程等学科提供工作空间.

Both the real-variable theory of function spaces and the boundedness of operators have always been one of the core subjects of harmonic analysis. This project, based on the existing works of the applicant and his collaborators about the real-variable theory of function spaces and the boundedness of operators on both variable and Musielak-Orlicz-type function spaces and so on, via thoroughly revealing the analytical representations of the geometric properties of the underlying spaces, skillfully combining them with the fine structures of basic function spaces, and applying some basic theories and methods from harmonic analysis, functional analysis and other related analysis subjects, aims to obtain some necessarily fundamental tools to develop a complete real-variable theory of function spaces; from this, this project then aims to develop a complete real-variable theory, including their various fine real-variable characterizations, of function spaces based on some basic function spaces such as generalized mixed-norms (variable) function spaces and Musielak-Orlicz-slice spaces over spaces of homogeneous type or other underlying spaces; applying these real-variable characterizations, this project aims to further establish the bilinear decomposition of the product of corresponding Hardy-type spaces and their dual spaces, from which this project aims then to study the boundedness of commutators, the div-curl lemma, general Leibniz-type rules and so on. All of these will enrich the real-variable theory of function spaces with fine structures, and provide working space for harmonic analysis, partial differential equations and other analytical subjects.