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

该奖项支持数学研究的继续 与全纯函数和映射有关的问题 多复变量理论中的一个问题 除了 函数论的方法，深入探讨了 偏微分方程的数学模型 在几个复杂的开放性问题中， 变量是理解的正交投影 函数（具有有限二次范数）定义在域中 多复变量空间的子空间 全纯函数 这被称为伯格曼投影。 主要目标是确定投影何时保持 可微性到边界。 有许多解决办法， 在各种有限性条件下， 域 目前的工作将遵循调查的路线 涉及到特殊向量场的构造， to a broad广泛class类of results结果. 最终目标是利用这些 建立整环正则性一般理论的技巧 无限类型。 Bergman投影的连续性和 非二次空间中相关d-杆Neumann算子 Sobolev空间也将被研究。 由于伯格曼投影并不总是规则的， 需要确定对规律性的障碍， 其存在的条件。 已知的是， 无穷型点集不足以给出 有关障碍物的完整信息。 一些潜在的- 必须找到d-杆Neumann算子的理论条件 与任何拓扑条件进行联合收割机。

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.