On compactness and $L^p$-regularity in the $\overline{\partial }$-Neumann problem

Abstract

Let $\Omega$ be a $C^4$-smooth bounded pseudoconvex domain in ${\mathbb{C}}^2$. We show that if the $\overline{\partial }$-Neumann operator $N_1$ is compact on $L_{(0,1)}^2(\Omega )$ then the embedding operator $\mathcal{J}:Dom(\overline{\partial })\cap Dom({\overline{\partial }}^{\ast })\to L_{(0,1)}^2(\Omega )$ is $L^p$-regular for all $2\le p<\infty$.

Minor changes. To appear in Bull. Lond. Math. Soc