Some Results for the Szegő and Bergman Projections on Planar Domains

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 $p=1,\mathrm{\infty }$ for domains with sufficient smoothness. We also pose some open questions that naturally arise from our investigation.