A Békollè-Bonami Class of Weights for Certain Pseudoconvex Domains

Abstract

We prove the weighted L p regularity of the ordinary Bergman projection on certain pseudoconvex domains where the weight belongs to an appropriate generalization of the Bekollé-Bonami class. The main tools used are estimates on the Bergman kernel obtained by McNeal and Bekollé's original approach of proving a good-lambda inequality.

A survey of the $$L^p$$ regularity of the Bergman projection

“…The estimates (2.3) are sometimes referred as the Bekollé-Bonami estimates. We also refer to [HWW20b,HWW20a] for a generalization on pseudoconvex domains. This chain of equivalences implies that smoothness of the conformal map F on the closure of the domains determines the regularity of the Bergman kernel and therefore also the estimates on the projection operator.…”

“…Furthermore, relate the operator norm of B Ω to the weight σ. Recently, [HWW20b] and [HWW20a] answered this question on some pseudoconvex domains on which sharp off-diagonal estimates on the Bergman kernel are known. They use a careful construction of dyadic decomposition on these domains by generalizing Carleson tents on the unit disc.…”

Bergman projection on the symmetrized bidisk

“…Note that there are extensive recent studies on the Bekollé-Bonami estimates (cf. [HW,HWW1,HWW2] and the references therein).…”