On compactness of the dbar-Neumann problem and Hankel operators

TL;DR

This paper investigates the relationship between the compactness of the ar-Neumann operator and Hankel operators on certain complex domains, revealing conditions under which one is compact but not the other.

Contribution

It demonstrates that on specific pseudoconvex domains, the Hankel operator can be compact while the ar-Neumann operator is not, highlighting nuanced differences in their properties.

Findings

Hankel operator is compact for all continuous symbols on the domain.

ar-Neumann operator is not compact despite Hankel operator's compactness.

The domain's boundary geometry influences the compactness properties of these operators.

Abstract

Let $\D=\D_1\setminus \Dc_2$, where $\D_1$ and $\D_2$ are two smooth bounded pseudoconvex domains in $\C^n, n\geq 3,$ such that $\Dc_2\subset \D_1.$ Assume that the $\dbar$-Neumann operator of $\D_1$ is compact and the interior of the Levi-flat points in the boundary of $\D_2$ is not empty (in the relative topology). Then we show that the Hankel operator on $\D$ with symbol $\phi, H^{\D}_{\phi},$ is compact for every $\phi\in C(\Dc)$ but the $\dbar$-Neumann operator on $\D$ is not compact.