Mathematical Sciences: Harmonic Analysis on Domains

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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. ***