Bekollé-Bonami estimates on some pseudoconvex domains

Abstract

We establish a weighted L p norm estimate for the Bergman projection for a class of pseudoconvex domains. We obtain an upper bound for the weighted L p norm when the domain is, for example, a bounded smooth strictly pseudoconvex domain, a pseudoconvex domain of finite type in C 2 , a convex domain of finite type in C n , or a decoupled domain of finite type in C n . The upper bound is related to the Bekollé-Bonami constant and is sharp. When the domain is smooth, bounded, and strictly pseudoconvex, we also obtain a lower bound for the weighted norm.

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).…”