Weighted estimates for the Bergman projection on planar domains

TL;DR

This paper studies weighted Lebesgue space estimates for the Bergman projection on planar domains, linking the estimates to the domain's Riemann map regularity, and extends known bounds with new necessary conditions and weak-type estimates.

Contribution

It introduces a regularity condition on the Riemann map necessary for full weighted estimates and strengthens conditions to achieve weak-type (1,1) bounds for the Bergman projection.

Findings

Established necessary regularity conditions for weighted estimates.

Extended bounds to include weak-type (1,1) estimates.

Linked conformal mapping properties with harmonic analysis techniques.

Abstract

We investigate weighted Lebesgue space estimates for the Bergman projection on a simply connected planar domain via the domain's Riemann map. We extend the bounds which follow from a standard change-of-variable argument in two ways. First, we provide a regularity condition on the Riemann map, which turns out to be necessary in the case of uniform domains, in order to obtain the full range of weighted estimates for the Bergman projection for weights in a B\'{e}koll\`{e}-Bonami-type class. Second, by slightly strengthening our condition on the Riemann map, we obtain the weighted weak-type $(1,1)$ estimate as well. Our proofs draw on techniques from both conformal mapping and dyadic harmonic analysis.