On compactness of the $\bar{\partial }$-Neumann operator on Hartogs domains

Abstract

We show that Property $(P)$ of $\partial \Omega$, compactness of the $\bar{\partial }$-Neumann operators $N_1$, and compactness of Hankel operator on a smooth bounded pseudoconvex Hartogs domain $\Omega =\{(z,w_1,w_2,\dots ,w_n)\in {\mathbb{C}}^{n+1}\mid {\sum }_{k=1}^n|w_k{|}^2<e^{-2\phi (z)},z\in D\}$ are equivalent, where $D$ is a smooth bounded connected open set in $\mathbb{C}$.