Analysis Seminar: "Irregularities of the Bergman projection and substitute operators"

Speaker: Luke Edholm, University of Michigan

Abstract: Let $\Omega \subset \mathbb{C}^n$ be a domain and define the Bergman space $A^p(\Omega)$ to be the space of holomorphic $L^p$ functions on $\Omega$.  Established L^p mapping properties of the Bergman projection known to hold on large classes of smooth, bounded, pseudoconvex domains are shown to fail for a family of non-smooth, bounded, pseudoconvex domains.  This presents challenges to the development of holomorphic approximation theorems and to the classification of dual spaces.  Despite these challenges, new substitute operators are constructed which avoid the deficiencies present in the original Bergman projection. These operators allow for concrete statements of approximation and duality theorems in this setting.  This work is joint with Debraj Chakrabarti and Jeff McNeal.

Host: Brett Wick