Sobolev regularity of the Bergman projection on certain pseudoconvex domains

Abstract

In this paper we study the Sobolev regularity of the Bergman projection $B$ and the $\overline{\partial }$-Neumann operator $N$ on a certain pseudoconvex domain. We show that if $\Omega$ is a domain with Lipschitz boundary, which is relatively compact in an $n$-dimensional compact Kähler manifold and satisfies some “$log\delta$-pseudoconvexity” condition, the operators $B$, $N$ and ${\overline{\partial }}^{\ast }N$ are regular in the Sobolev spaces $W_{r,s}^k(\Omega ,E)$ for forms with values in a holomorphic vector bundle $E$ and for any $k<\eta /2$, $0<\eta <1$, $0\le r\le n$, $0\le s\le n-1$.