Schatten class Hankel and $\overline{\partial }$-Neumann operators on pseudoconvex domains in ${\mathbb{C}}^n$

Abstract

Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in ${\mathbb{C}}^n$ for $n\ge 2$ and let $\phi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\phi }}$ is in Schatten $p$-class for $p\le 2n$ then $\phi$ is a constant function. As a corollary, we show that the $\overline{\partial }$-Neumann operator on $\Omega$ is not Hilbert-Schmidt.

minor changes, to appear in Monatsh. Math