广义Fock空间及相关积分算子

This project deals with generalized Fock spaces and related integral operators in several complex variables. Applying mordern Functional Analysis, Differential Geometry and PDE, we will study the pointwise estimates for Bergman kernel on generalized Fock spaces and the related Bergman metric. We will focus on the atomic decomposition and the interpolation for generalized Fock spaces, characterize these Fock space functions via their partial derivatives and radial derivatives. Moreover, we will obtain the characterizations on some integral operators, such as extended Cesaro operators and Toeplitz operators, on generalized Fock spaces, including boundedness, compactness, Schatten classes, Schatten-Herz classes and invertibility properties.

本项目以多复变数广义Fock空间及其上相关积分算子作为研究对象。应用泛函分析、微分几何和偏微分方程等现代数学工具，建立多复变广义Fock空间上Bergman核的点态估计，并以此为基础来讨论与Bergman度量相应的几何特征刻画；研究广义Fock空间的原子分解定理和插值定理，并借助梯度、径向导数、切向导数等刻画该空间的范数等价性；获得广义Fock空间上Toeplitz算子和广义Cesaro算子等积分算子的有界性、紧性、Schatten类、Schatten-Herz类和可逆性特征。

本项目主要以多复变数广义Fock空间及其上相关积分算子作为研究对象。应用泛函分析、微分几何和偏微分方程等现代数学工具，建立了广义Fock空间和指数型Bergman空间上Bergman核的点态估计，并以此为基础来讨论与Bergman度量相应的几何特征刻画；获得了这两类空间上Carleson测度的性质刻画、Bergman投影的特性、稠子集和共轭空间等性质；定义了一类比BMO空间更一般化的IMO空间；得到了该空间的分解定理和范数等价刻画；研究了广义Fock空间和Bergman空间上Hankel算子、小Hankel算子、Toeplitz算子和弱局部算子等积分算子的有界性、紧性和Schatten类等特征。