Mathematical Sciences: Function Theory on Convex Domains

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The project is concerned with mathematical research on the function theory on domains in the space of several complex variables. Fundamental ideas from complex analysis, partial differential equations, operator theory and integral representations are combined in these investigations. One basic goal - well understood in one variable but open in several variables - is the classification of domains into equivalence classes under holomorphic coordinate changes. Considerable progress has been made recently in this area through research on related problems of extending holomorphic maps to the boundary of regions. This was done by analysis of the Bergman projection operator. This focused attention on projections of functions onto holomorphic function spaces such as the Bergman and the Szego projections. Work will be done analyzing the regularity properties of these and related projections on convex domains. Corvex domains with real analytic boundary are of finite type so that well known methods of partial differential equations can be used to show that these projection operators preserve differentiability of functions in Sobolev spaces. Work will be done in expanding present methods and developing new tools to handle domains of infinite type.