Mathematical Sciences: Function Theory on Convex Domains

该项目是关于几个复变量空间中区域上的函数论的数学研究。这些研究结合了复分析、偏微分方程、算子论和积分表示的基本思想。一个基本的目标--在一个变量中被很好地理解，但在多个变量中是开放的--是在全纯坐标变化下将域分类为等价类。近年来，通过将全纯映射推广到区域边界的相关问题的研究，在这一领域取得了相当大的进展。这是通过对Bergman投影算子的分析来完成的。这将注意力集中在函数在全纯函数空间上的投影上，如Bergman投影和Szego投影。我们将分析这些投影和相关凸域上投影的正则性。具有实解析边界的Corvex区域是有限类型的，因此众所周知的偏微分方程方法可以用来证明这些投影算子在Sobolev空间中保持函数的可微性。将在扩展现有方法和开发处理无限类型域的新工具方面进行工作。

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.