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

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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.