On compactness of the $\overline {\partial }$-Neumann problem and Hankel operators

Abstract:

Let $\Omega =\Omega _1\setminus \overline {\Omega }_2$, where $\Omega _1$ and $\Omega _2$ are two smooth bounded pseudoconvex domains in $\mathbb {C}^n, n\geq 3,$ such that $\overline {\Omega }_2\subset \Omega _1.$ Assume that the $\overline {\partial }$-Neumann operator of $\Omega _1$ is compact and the interior of the Levi-flat points in the boundary of $\Omega _2$ is not empty (in the relative topology). Then we show that the Hankel operator on $\Omega$ with symbol $\phi , H^{\Omega }_{\phi },$ is compact for every $\phi \in C(\overline {\Omega })$ but the $\overline {\partial }$-Neumann operator on $\Omega$ is not compact.