Compactness of the canonical solution operator on Lipschitz q-pseudoconvex boundaries

Abstract

Let $\Omega\subset\mathbb{C}^n$ be a bounded Lipschitz q-pseudoconvex domain that admit good weight functions. We shall prove that the canonical solution operator for the $\overline{\partial}$-equation is compact on the boundary of $\Omega$ and is bounded in the Sobolev space $W^k_{r,s}(\Omega)$ for some values of $k$. Moreover, we show that the Bergman projection and the $\overline\partial$-Neumann operator are bounded in the Sobolev space $W^k_{r,s}(\Omega)$ for some values of k. If $\Omega$ is smooth, we shall give sufficient conditions for compactness of the $\overline\partial$-Neumann operator.

Keywords

• Lipschitz domain
• q-pseudoconvex domain