本项目以多复变数Julia引理及其应用、高维Fock空间及相关算子为研究对象，拟用泛函分析、微分几何、李代数和PDE等现代数学工具，建立C^n中单位球、单位多圆柱、Rienhardt域和强凸域等上的Julia引理，以多复变数Julia引理为工具来研究某些全纯映射族的Bloch常数，刻画多复变数Fock空间的结构特征以及其上Hankel算子的有界性、紧性和Schatten类等；将多复变的观点与成果应用到调和分析，刻画高维调和Fock空间的Bergman核与Bergman度量的特性，研究高维调和Fock空间的Carleson测度、Berezin变换与Bergman型积分算子的性质以及在该空间上Toeplitz算子与Hankel算子的特性。. 本项目以单复变中的一些经典问题和热点问题为出发点，将研究视线转向多复变数，并致力于在调和分析中的应用，力求在学科方向的交叉研究上寻找新的生长点。

This project deals with the Julia lemma in several complex variables and its application, and with the high dimensional Fock space and the related operators on this space. Applying modern Functional Analysis, Differential Geometry, Lie Algebra and PDE, we will establish the Julia lemmas on a variety of domains of C^n including the unit ball, unit polydisk, Rienhardt domain and strongly convex domain, and apply the Julia lemma to obtain the sharp estimates of the Bloch constant for some holomorphic mappings. We will characterize the properties of the Fock space in several complex variables, and discuss the boundedness, compactness and Schatten class of the Hankel operator on this space. Moreover, we will use the ideals and results of several complex variables to harmonic analysis, and study the behavior of the Bergman kernel and the related Bergman metric on the high dimensional harmonic Fock space. We will focus on the Carleson measure, Berezin transform and Bergman-type integral operator on the high dimensional harmonic Fock space. Finally, we will study the Toeplitz operator and Hankel operator on this space as well. . Taking the classical problems and hot problems in complex analysis as the starting point, we consider these problems in several complex variables, and devote to the application in harmonic analysis. We try to find a new growth point in the crossover study. Some new ideas and tools are developed, which, in our belief, have their own interests in other aspects.