Mathematical Sciences: Harmonic Analysis on Domains

小行星9400772 该奖项支持在两个相关领域出现的问题的数学研究。 第一个是关于研究的霍奇理论的de Rham复杂使用不同的内积，通常的二次之一。 这项工作应提供新的见解的规律性理论的椭圆问题以及洞察结构的谐波空间。 德·拉姆复形的霍奇理论具有重要的几何解释。 第二个项目涉及特殊功能空间的发展，以适应特定领域的特定边界值问题。 沿着算子理论的最新进展，为特定应用设计函数的思想应该具有深远的应用。 在另一个方向上，工作将在调和分析和函数理论的基本要素在复欧几里德空间域上。 这包括理论的哈代空间，功能的有界平均振荡，电晕问题，和Lipschitz空间。 在工作中的一个重要主题集中在几何边界的域，以及它如何决定函数理论的内部。 特别是问题的边界唯一性和刚性的全纯函数和映射将采取。 偏微分方程是物理世界数学模型的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 当方程定义在多个复变量的域上时，就像他们在这个项目中一样，工作的范围大大增加，以考虑到在研究多个复变量的全纯函数和相应的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. ***