Weighted Bergman spaces and the $\bar{\partial}-$equation

TL;DR

This paper develops $L^2$ estimates for the $ar{ ext{d}}$-equation on pseudoconvex domains with weighted measures, leading to new results in weighted Bergman space theory, including theorems on boundary behavior, density, and kernel relations.

Contribution

It introduces a H"ormander type $L^2$ estimate for the $ar{ ext{d}}$-equation with weighted measures on pseudoconvex domains, advancing the understanding of weighted Bergman spaces.

Findings

Established a corona type theorem for weighted Bergman spaces.

Proved an analogue of the Levi problem for $A^2_\alpha(\Omega)$.

Derived an optimal Gehring type estimate for functions in $A^2_\alpha(\Omega)$.

Abstract

We give a H\"ormander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $\delta_\Omega^{-\alpha}dV$, $\alpha<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function theory of weighed Bergman spaces $A^2_\alpha(\Omega)$ are given, including a corona type theorem, a Gleason type theorem, together with a density theorem. We investigate in particular the boundary behavior of functions in $A^2_\alpha(\Omega)$ by proving an analogue of the Levi problem for $A^2_\alpha(\Omega)$ and giving an optimal Gehring type estimate for functions in $A^2_\alpha(\Omega)$. A vanishing theorem for $A^2_1(\Omega)$ is established for arbitrary bounded domains. Relations between the weighted Bergman kernel and the Szeg\"o kernel are also discussed.