Some Results for the Szeg\H{o} and Bergman Projections on Planar Domains

TL;DR

This paper establishes boundedness, compactness, and weighted estimates for Bergman and Szeg\

Contribution

It provides new quantitative $L^p$ estimates and analyzes the difference between projections on smooth planar domains.

Findings

Weighted estimates for projections

Compactness at $p=1,\infty$ for smooth domains

Explicit $p$-range estimates on specific domains

Abstract

The purpose of this note is to prove some boundedness/compactness results of a harmonic analysis flavor for the Bergman and Szeg\H{o} 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 Szeg\H{o} 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.