PhD Thesis Defense: "Weighted Estimates for the Bergman and Szego Projections"

Abstract: The Bergman and Szego Projections are orthogonal projections onto spaces of holomorphic functions and are fundamental objects in complex analysis. Although the L^2 boundedness of these projections is trivial, the continuity of the operators on L^p spaces, including weighted spaces, has been a topic of significant interest in the research community. In this thesis, we undertake a study of weighted estimates for these projection operators on large classes of pseudoconvex domains. In particular, we prove weighted L^p estimates for the Bergman projection on several classes of smoothly bounded, pseudoconvex domains of finite type, we obtain weighted L^p estimates for the Bergman and Szego projections on strongly pseudoconvex domains with near-minimal smoothness, and we prove weighted endpoint estimates for both projection operators for similar domain classes. A major theme in this dissertation is the use of real-variable harmonic analysis tools to handle such questions. 

Host: Brett Wick