本项目的研究对象为C^N中单位球以及多圆柱上的全纯自映射所构成的单参量连续半群以及几类经典函数空间上的单参量复合算子半群和单参量加权复合算子半群。将考察单位球和多圆柱上全纯自映射的Schröder方程以及嵌入问题，全纯映射单参量半群的无穷生成元的参数表示、边界表现、刚性定理，全纯映射单参量半群的渐近表现、刚性定理，全纯映射单参量半群对应的上边缘、闭上链，单参量复合算子半群以及单参量加权复合算子半群在经典的函数空间(比如Hardy空间、BMOA空间、VMOA空间、Bloch空间等)上的强连续性、极大强连续子空间等问题。

Our program focus on one-parameter continuous semigroups of holomorphic self-maps on the unit ball and the polydisk of C^N and one-parameter continuous semigroups of composition operators and one-parameter continuous semigroups of weighted composition operators on several classical function spaces on the unit ball and the polydisk. We will solve the Schröder’s equations of holomorphic self-maps on the unit ball and the polydisk, and give conditions for when a self-map could be embedded into a semigroup. Boundary behaviors, rigidity theorems and parametric representations of infinitesimal generators of semigroups will be considered. Properties of holomorphic one-parameter continuous semigroups, such as asymptotic behavior, rigidity, are goals of our program too. For a given one-parameter continuous semigroup, we will give a survey on the relationships of the corresponding coboundries and cocycles. We will try to find out what kinds of one-parameter continuous composition and weighted composition operator semigroups are strong continuous on classical function spaces, such as Hardy space, BMOA, VMOA and Bloch space. Maximal strong continuous spaces of composition and weighted composition operator semigroups will be computed.