多复变光滑拟凸Hartogs域上Dbar-Neumann算子的紧性研究

本项目以多复变中的Dbar-Neumann算子Nq为研究对象，运用函数分析和椭圆偏微分方程理论，研究该算子在高维复空间中光滑拟凸Hartogs域上的L²-紧性。申请者前期研究发现一些使得弱拟凸域上Nq具有L²-紧性的充分条件，为Nq紧性的研究带来了新的技术思路。获得该类充分条件的方法是基于对经典L²加权估算式的改良并将其与边界无限型点的分析性质相联系。项目的核心目标是在一般高维复空间中，研究光滑拟凸Hartogs域上Nq具有L²-紧性的充分必要条件。项目有如下研究内容：1)、探索上述区域上Nq具有紧性的必要条件；2)、寻找上述区域上其他蕴含Nq紧性的充分条件；3)、分析上述区域上Nq具有紧性与边界无限型点的相应几何分析性质的关联。项目的研究成果有助于进一步探索Dbar-Neumann算子在一般弱拟凸域上具有紧性的充要条件，并能启发全纯映射延拓性质的研究以及拟凸域几何性质的研究。

In this project, the main topic is to study L²-compactness of the Dbar-Neumann operator Nq on smooth pseudoconvex Hartogs domains in high-dimensional complex spaces, by using analysis and elliptic partial differential equation theory. The applicant established several new sufficient conditions for L²-compactness of Nq, which brought new ideas to the research of compactness of Nq. These new results are based on the improvement of classical weighted L²-estimate and its relation with analysis properties of the infinite type points in the boundary. The main research goal of this project is to study necessary and sufficient conditions for compactness of the Dbar-Neumann operator Nq on smooth pseudoconvex Hartogs domains in Cⁿ. The applicant will study the following topics: 1) necessary conditions for compactness of the Dbar-Neumann operator on the above domains; 2) new sufficient conditions for compactness of the Dbar-Neumann operator on the above domains; 3) relations between compactness of the Dbar-Neumann operator and geometric analysis conditions of the boundary. The result of this project will stimulate the study of L²-compactness of Nq on general weakly pseudoconvex domains, the extension theory of holomorphic mappings, and the geometry theory on weakly pseudoconvex domains.