Analysis Seminar/Third Year Candidacy Requirement: "Weighted estimates for the Bergman and Szegö projections"

Abstract: The Bergman and Szegö Projections on a domain and the boundary of a domain, respectively, are important operators in complex analysis and function theory. While they are automatically bounded on L^2, it is natural to consider them as integral operators and study their regularity on L^p. We review some results in the literature concerning the L^p boundedness of these operators on various types of domains. In particular, we will discuss the important technique of relating these operators to well-known integral representations of holomorphic functions, and how this leads to a proof of L^p regularity in the case the domain is strongly pseudoconvex with minimal smoothness. We will also review Bekolle's characterization of the weights w for which the Bergman projection is bounded on L^p(w) on the unit ball. At the end of the talk, we will discuss some recent and potential results concerning weighted estimates of these operators on more general domains.

Host: Brett Wick