多复变数全纯映射与函数空间理论若干前沿问题的研究

多复变函数论是现代数学的主流方向之一.本项目以多复变数的Schwarz引理、逆紧全纯映射的刚性、Campanato空间和Toeplitz算子等为研究对象，拟用分析、代数、几何等现代数学工具，建立高维复空间中某些区域在特征流形处的Schwarz引理；利用CR几何、正规型理论、次椭圆分析等技术，研究推广球之间逆紧全纯映射的刚性问题；刻画单位球上Campanato空间的Carleson测度和Poisson-Szegö核，解决Campanato空间的Gleason问题，获得Campanato空间上Toeplitz算子的性质；通过超几何函数、Cayley变换、Schur检验等方法，得到上半平面或Siegel上半空间上Campanato空间的结构特征，研究其上相关算子的特性.本项目聚焦多复变的若干前沿问题和难点问题，研究内容新且深刻，学科交叉性和应用性强，有望取得突破或显著进展，并形成新的方法和理论.

Function theory in several complex variables is one of the main directions in modern mathematics. This project deals with Schwarz lemma, the rigidity of proper holomorphic mapping, Campanato space and Toeplitz operator in several complex variables. Applying Analysis, Algebra, Geometry and other modern mathematical tools, we will establish some Schwarz lemmas at the characteristic manifold of some domains in high-dimensional complex space. With CR geometry, normal form theory and sub-ellipse analysis, we will study the rigidity problem for proper holomorphic mappings between the generalized balls. We will characterize Carleson measure and Poisson-Szegö kernel, solve Gleason problem and obtain some properties of the Toeplitz operator for the Campanato space on the unit ball. In the setting of the upper half plane or the Siegel upper half space, by using the hypergeometric function, Cayley transform and Schur test, we will give some structure characteristics of the Campanato space, and study the properties of the related linear operators on the Campanato space. We aim at the research and exploration of some frontier problems and difficult problems in this field, and expect to make breakthroughs or significant progress. Some new ideas and tools are developed, which, in our belief, have their own interests in other aspects.