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

TL;DR

This paper establishes the equivalence of several compactness properties related to the $ar{ ext{d}}$-Neumann operator on smooth Hartogs domains, linking geometric and operator-theoretic conditions.

Contribution

It proves the equivalence between Property (P), compactness of the $ar{ ext{d}}$-Neumann operator, and compactness of Hankel operators on Hartogs domains.

Findings

Property (P) holds if and only if the $ar{ ext{d}}$-Neumann operator is compact.

Compactness of Hankel operators is equivalent to the other properties.

The results connect geometric boundary conditions with operator compactness in complex analysis.

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 $\Omega={\{(z, w_1, w_2,\dots, w_n) \in \mathbb{C}^{n+1} \mid\sum_{k=1}^{n} |w_k|^2 < e^{-2\varphi(z)}, z\in\mathit{D}\}}$ are equivalent, where $D$ is a smooth bounded connected open set in $\mathbb{C}$.