Some Results for the Szegő and Bergman Projections on Planar Domains

Abstract

The purpose of this note is to prove some boundedness/compactness results of a harmonic analysis flavor for the Bergman and Szegő projections on certain classes of planar domains using conformal mappings. In particular, we prove weighted estimates for the projections, provide quantitative $L^p$ estimates and a specific example of such estimates on a domain with a sharp $p$ range, and show that the ``difference'' of the Bergman and Szegő projections is compact at the endpoints $p=1,\infty$ for domains with sufficient smoothness. We also pose some open questions that naturally arise from our investigation.

arXiv comment17 pages. Comments or suggestions welcome. To appear in Contemporary Mathematics Conference Proceedings