Mathematical Sciences: Harmonic Analysis on Domains

9400772《将军》这个奖项支持对两个相关领域中出现的问题进行数学研究。第一个是关于用不同于通常的二次型内积的De Rham复数的Hodge理论的研究。这项工作将为椭圆问题的正则性理论提供新的见解，并为调和空间的结构提供新的见解。德罗姆复合体的霍奇理论有重要的几何解释。第二个项目涉及开发适应于特定区域上的特定边值问题的特殊函数空间。随着算子理论的最新进展，为特定应用设计函数的想法应该具有深远的应用。在另一个方向，工作将是关于复欧氏空间中区域的调和分析和函数论的基本元素。这包括Hardy空间理论、有界平均振荡函数、日冕问题和Lipschitz空间。这部作品的一个重要主题集中在区域边界的几何以及它如何决定内部的函数理论。特别地，将讨论全纯函数和映射的边界唯一性和刚性问题。偏微分方程构成了对物理世界进行数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。当方程被定义在多个复变量的区域上时，就像在这个项目中一样，工作的范围大大增加，以考虑在研究几个复变量的全纯函数和相应的区域中现有的大量工作。***

9400772 Krantz This award supports mathematical research on problems arising in two related areas. The first concerns the study of Hodge theory for the de Rham complex using different inner products that the usual quadratic one. The work should provide new insights into the regularity theory for elliptic problems as well as insight into the structure of the harmonic space. The Hodge theory for the de Rham complex has important geometric interpretations. A second project involves the development of special function spaces adapted to specific boundary value problems on specific domains. Along with recent advances in operator theory, the idea of designing functions for particular applications should have far-reaching applications. In another direction, work will be don on basic elements of harmonic analysis and function theory on domains in complex Euclidean space. This includes the theory of hardy spaces, functions of bounded mean oscillation, the corona problem, and Lipschitz spaces. An important theme in the work focuses on the geometry of the boundary of the domain and how it determines the function theory on the interior. In particular questions of boundary uniqueness and rigidity for holomorphic functions and mappings will be taken up. Partial differential equations form a basis for mathematicalmodeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. When the equations are defined over domains in several complex variables, as they are in this project, the scope of the work increases considerably to take into account the large body of existing work available in the study of holomorphic functions of seve ral complex variables and the corresponding domains. ***