Mathematical Sciences: The Bergmann Projection, the d-bar- Neumann Operator, and Holomorphic Mappings

数学科学：伯格曼投影、d-bar-诺依曼算子和全纯映射

• 批准号: 9203514
• 负责人: Harold Boas
• 金额: $ 3万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1994-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203514&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Bergmann; Projection; bar

This award supports a continuation of mathematical research into problems associated with holomorphic functions and mappings in the theory of several complex variables. In addition to function theoretic methods, the work exploits the deep connections of the subject with partial differential equations. Among the most challenging open questions in several complex variables is that of understanding the orthogonal projection of functions (with finite quadratic norm) defined on a domain in the space of several complex variables onto the subspace of holomorphic functions. This is known as the Bergman projection. The primary goal is to determine when the projection preserves differentiability up to the boundary. A number of solutions have been determined under various finiteness conditions on the domain. The current work will pursue a line of investigation involving the construction of special vector fields which has led to a broad class of results. The ultimate goal is to use these techniques to establish a general theory of regularity on domains of infinite type. The continuity of the Bergman projection and the related d-bar Neumann operator in spaces other than quadratic Sobolev spaces will also be studied. Since the Bergman projections are not always regular, there is a need to determine obstructions to regularity as well as conditions for its presence. It is known that the topology of the set of points of infinite type is not sufficient to give complete information about obstructions. Some potential- theoretic condition on the d-bar Neumann operator must be found to combine with any topological conditions.

该奖项支持继续对与全纯函数和多个复变量理论中的映射相关的问题进行数学研究。 除了函数论方法之外，这项工作还利用了该主题与偏微分方程的深层联系。 多个复变量中最具挑战性的开放问题之一是理解在多个复变量空间中的域上定义的函数（具有有限二次范数）到全纯函数子空间的正交投影。 这称为伯格曼投影。 主要目标是确定投影何时保留直至边界的可微性。 在域上的各种有限性条件下已经确定了许多解。 目前的工作将进行一系列涉及特殊矢量场构建的调查，这已经产生了广泛的结果。 最终目标是使用这些技术建立无限类型域上的一般规律理论。 还将研究二次 Sobolev 空间以外的空间中 Bergman 投影和相关 d-bar Neumann 算子的连续性。 由于伯格曼预测并不总是规则的，因此需要确定规则性的障碍及其存在的条件。 众所周知，无限类型点集的拓扑不足以给出有关障碍物的完整信息。 必须找到与任何拓扑条件相结合的 d-bar Neumann 算子的一些势理论条件。