Bergman projection induced by kernel with integral representation

TL;DR

This paper characterizes bounded Bergman projections induced by kernels with integral representations and explores related inequalities using Bekollé-Bonami and Sawyer-testing conditions.

Contribution

It introduces new characterizations of boundedness for Bergman projections with specific kernels and analyzes associated inequalities using advanced weight conditions.

Findings

Characterization of bounded Bergman projections via Bekollé-Bonami conditions.

Two-weight inequalities for maximal Bergman projections using Sawyer-testing.

Conditions for (1,1)-inequality in the context of integral kernel representations.

Abstract

Bounded Bergman projections $P_\omega:L^p_\omega(v)\to L^p_\omega(v)$, induced by reproducing kernels admitting the representation $$ \frac{1}{(1-\overline{z}\zeta)^\gamma}\int_0^1\frac{d\nu(r)}{1-r\overline{z}\zeta}, $$ and the corresponding (1,1)-inequality are characterized in terms of Bekoll\'e-Bonami-type conditions. The two-weight inequality for the maximal Bergman projection $P^+_\omega:L^p_\omega(u)\to L^p_\omega(v)$ in terms of Sawyer-testing conditions is also discussed.