全纯（解析）函数空间理论与许多学科有着密切的联系, 在现代数学中起着非常重要的作用，如在调和分析、偏微分方程与算子理论领域。 本项目将重点研究几类重要的复函数空间，包括Hardy空间、Dirichlet空间、Bergman空间、 Morrey空间等一类全纯函数空间性质，主要关注在多变量下的复函数空间的循环元问题、Bergman空间的内函数与外函数的特征, Bergman空间的Korenblum最大模原理和在有限型凸域上几类函数空间的Corona型分解问题。 同时也给出多种算子在几类新型复函数空间上的有界性、紧性的刻划及其范数和本性范数的大小估计。

Theory of the holomorphic （analytic） function spaces are closely related to many areas of mathematics. It plays very important roles in harmonic analysis, partial differential equations and operator theory. The project focuses on some important holomorphic function spaces such that Hardy spaces、 Dirichlet spaces、Bergman spaces and Morrey spaces. We study the characteristic problems of cyclic elements on several holomorphic function spaces, the characteristic of inner functions and outer functions in Bergman spaces, Korenblum’s maximum principle problems in Bergman spaces and Corona problems on those spaces over finite type convex domains. Also, we will characterize the boundedness and compactness of some operators on some new function spaces, and calculate the norm and essential norm of those operators.