Mathematical Sciences: Function Theory on Convex Domains

本课题是关于复变空间中域上的函数理论的数学研究。这些研究结合了复分析、偏微分方程、算子理论和积分表示的基本思想。一个基本目标是在全纯坐标变化下将域划分为等价类，这个目标在一个变量中很容易理解，但在几个变量中却很开放。近年来，全纯映射扩展到区域边界的相关问题的研究取得了相当大的进展。这是通过分析伯格曼投影算子实现的。这集中于函数在全纯函数空间上的投影，如Bergman投影和Szego投影。工作将完成分析这些和凸域上相关投影的正则性。具有实解析边界的凸域是有限型的，因此可以用已知的偏微分方程方法来证明这些投影算子在Sobolev空间中保持函数的可微性。工作将在扩展现有的方法和开发新的工具来处理无限类型的域。

The project is concerned with mathematical research on the function theory on domains in the space of several complex variables. Fundamental ideas from complex analysis, partial differential equations, operator theory and integral representations are combined in these investigations. One basic goal - well understood in one variable but open in several variables - is the classification of domains into equivalence classes under holomorphic coordinate changes. Considerable progress has been made recently in this area through research on related problems of extending holomorphic maps to the boundary of regions. This was done by analysis of the Bergman projection operator. This focused attention on projections of functions onto holomorphic function spaces such as the Bergman and the Szego projections. Work will be done analyzing the regularity properties of these and related projections on convex domains. Corvex domains with real analytic boundary are of finite type so that well known methods of partial differential equations can be used to show that these projection operators preserve differentiability of functions in Sobolev spaces. Work will be done in expanding present methods and developing new tools to handle domains of infinite type.