\(L^p\) regularity of weighted Bergman projections (Q2838112)

This article investigates \(L^p\) regularity of weighted Bergman projections on the unit disc and of ordinary Bergman projections on related domains in \(\mathbb{C}^2\). The author obtains a class of radial weight functions on the unit disc for which the corresponding weighted Bergman projections are unbounded on the weighted \(L^p\) spaces when \(p \neq 2\). These weights lead to Reinhardt domains in \(\mathbb{C}^2\) for which the ordinary Bergman projections are bounded on \(L^p\) only when \(p=2\). Additionally the author constructs, for an arbitrary positive number~\(p_0\) larger than~\(2\), a weight on the unit disc for which the corresponding weighted Bergman projection is bounded on the weighted \(L^p\) space only when \(q_0<p<p_0\), where \(q_0^{-1}+p_0^{-1}=1\). Corresponding to such a weight is a Hartogs domain in \(\mathbb{C}^2\) for which the ordinary Bergman projection is bounded on \(L^p\) only when \(q_0<p<p_0\). The article is based on a part of the author's 2010 PhD dissertation directed by J.~D. McNeal.