本项目以多复变数加权Fock空间及其上相关积分算子为研究对象,强调全纯函数空间理论同泛函分析、偏微分方程和微分几何等现代数学分支的交叉融合.应用这些数学分支中的工具研究多复变数加权Fock空间上与Bergman度量相应的几何性质、空间范数等价表示、原子分解定理和插值定理等.合理定义Bergman度量下的函数空间BMO和IMO,研究以BMO函数为符号的Toeplitz算子在不同幂次加权Fock空间之间的有界性和紧性等泛函特征.研究Hankel算子从p次加权Fock空间到相应的q次Lebesgue空间之间的映射性质.

This project deals with weighted Fock spaces in several complex variables and related integral operators with the emphasis on the combination of holomorphic functions and Function Analysis, PDE and Differential Geometry. Using the tools in these branches of mathematics, we will study the geometric properties corresponding to Bergman metric on weighted Fock spaces. And we will focus on the equivalent norms, the atomic decomposition and the interpolation for weighted Fock spaces. We will define BMO and IMO spaces reasonably under related Bergman metric and characterize the mapping properties of Toeplitz operators with BMO symbols from the p-th weighted Fock space to the q-th one. Moreover, we will study the behavior of single Hankel operator mapping from the p-th weighted Fock space to the q-th Lebesgue space.