Kato平方根算子及分数阶微分算子的相关问题

调和分析是现代数学的重要组成部分，其发展过程与偏微分方程密切相关。本项目主要探讨调和分析与偏微分方程领域的若干交叉主题。特别地,我们将研究相关二阶散度型椭圆算子和Calderón交换子的一些重要的调和分析问题，其中包括建立与二阶散度型椭圆算子相关的Kato平方根、分数阶微分算子、分数次积分以及Littlewood-Paley算子生成的交换子的L^p理论，给出Calderón交换子及分数阶微分Calderón交换子在一些函数空间例如L^p空间、Morrey空间以及加权L^p空间的有界性特征刻画，建立带粗糙（变量）核的分数阶微分Calderón交换子的L^p理论。本项目的研究不仅是调和分析领域中算子理论、有界性特征刻画的自然延伸和发展，同时也将推动偏微分方程理论的研究。

Harmonic analysis is a very important branch of modern mathematics, the developing process of harmonic analysis is closely connected with the study of partial differential equation. This project is mainly devoted to study some interaction topics between harmonic analysis and PDE. Specially, this project will study some important harmonic analysis problems related to the second order divergence form elliptic operator. These include: establishing an L^p theory of the commutator of the Kato square root、fractional differential operator、fractional integral and Littlewood-Paley operator associated with the second order divergence form elliptic operator, giving some characterizations of the L^p boundedness、the Morrey spaces boundedness and the weighted L^p boundedness for the Calderón commutator and the fractional differential Calderón commutator, establishing the L^p theory for the Calderón commutator of the fractional differential operator with rough (variable) kernel. Furthermore, the study of this project is a natural development of the operator theory and the boundedness characterization in harmonia analysis as well as a promotion in investigating partial differential equations.