Approximation in weighted Bergman spaces and Hankel operators on strongly pseudoconvex domains

Abstract

Suppose D is a bounded strongly pseudoconvex domain in ${{\mathbb {C}}}^n$ with smooth boundary, and let $\rho$ be its defining function. For $1< p<\infty$ and $\alpha >-1$, we show that the weighted Bergman projection $P_\alpha$ is bounded on $L^p(D, |\rho |^\alpha dV)$. With non-isotropic estimates for $\overline{\partial }$ and Stein’s theorem on non-tangential maximal operators, we prove that bounded holomorphic functions are dense in the weighted Bergman space $A^p(D, |\rho |^\alpha dV)$, and hence Hankel operators can be well defined on these spaces. For all $1<p, q<\infty$ we characterize bounded (resp. compact) Hankel operators from p-th weighted Bergman space to q-th weighted Lebesgue space with possibly different weights. As a consequence, we generalize the main results in Pau et al. (Indiana Univ Math J 65:1639–1673, 2016) and resolve a question posed in Lv and Zhu (Integr Equ Oper Theory, 2019).