Mathematical Sciences: Bergman Spaces: Theory and Applications

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Korenblum 9302985 This project is a continuation of mathematical research into classical function theory focusing on Bergman spaces and developments which have issued from recent advances. A fundamental breakthrough occurred when H. Hedenmalm discovered a factorization of functions in Bergman space analogous to the classical one of F. Riesz which helped clarify the structure of the Hardy spaces. Among the goals of this research, work will be done to describe the invariant subspaces of the analytic Lebesgue spaces, on understanding the nonstandard invariant subspaces and determining the domination relations on zero sets. Work will also be done on the Green's function for the biharmonic operator in n-dimensional Euclidean space and algebras of harmonic functions in three-dimensional space. The latter is concerned with locating those complex harmonic functions whose modulus is also harmonic. These are thought to be the closest generalization to odd dimensions of analytic functions. Complex analysis has a long, distinguished mathematical history. It has been a basis for studies in diverse areas as number theory, computation, approximation theory and potential theory. Although the classical theory has been thoroughly investigated for centuries, the newer Bergman spaces have been slow to develop, despite their recognized importance. The recent work of Hedelmalm has completely changed the landscape to a point where one fully expects major thrusts to take place in this area similar to those which occurred when the theory of Hardy spaces flourished in the 1950's and 60's. ***