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

Abstract

We give a Hörmander 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 $A_{\alpha }^2(\Omega )$ 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 $A_{\alpha }^2(\Omega )$ by proving an analogue of the Levi problem for $A_{\alpha }^2(\Omega )$ and giving an optimal Gehring type estimate for functions in $A_{\alpha }^2(\Omega )$. A vanishing theorem for $A_1^2(\Omega )$ is established for arbitrary bounded domains. Relations between the weighted Bergman kernel and the Szegö kernel are also discussed.

23 pages; Some minor mistakes are corrected; to appear in Trans. AMS

References in corpus (1)

• A simple proof of the Ohsawa-Takegoshi extension theorem