Two weight inequality for Bergman projection

TL;DR

This paper characterizes the boundedness of the weighted Bergman projection on L^p spaces with two weights, using Muckenhoupt and Bekollé-Bonami conditions, and analyzes the asymptotic behavior of kernels.

Contribution

It provides a new characterization of the two weight inequality for Bergman projections via self-improving conditions under smoothness assumptions.

Findings

Boundedness characterized by Muckenhoupt and Bekollé-Bonami conditions.

Asymptotic analysis of L^p-means and kernel integrability.

Conditions depend on smoothness of weights.

Abstract

The motivation of this paper comes from the two weight inequality $$\|P_\omega(f)\|_{L^p_v}\le C\|f\|_{L^p_v},\quad f\in L^p_v,$$ for the Bergman projection $P_\omega$ in the unit disc. We show that the boundedness of $P_\omega$ on $L^p_v$ is characterized in terms of self-improving Muckenhoupt and Bekoll\'e-Bonami type conditions when the radial weights $v$ and $\omega$ admit certain smoothness. En route to the proof we describe the asymptotic behavior of the $L^p$-means and the $L^p_v$-integrability of the reproducing kernels of the weighted Bergman space $A^2_\omega$.