On compactness of Hankel and the $\bar{\partial}$-Neumann operators on Hartogs domains in $\mathbb{C}^2$

Abstract

We prove that on smooth bounded pseudoconvex Hartogs domains in $\mathbb{C}^2$ compactness of the $\bar{\partial}$-Neumann operator is equivalent to compactness of all Hankel operators with symbols smooth on the closure of the domain.