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

TL;DR

This paper establishes an equivalence between the compactness of the ar-Neumann operator and Hankel operators on smooth bounded pseudoconvex Hartogs domains in Cb2, linking two important operator properties.

Contribution

It proves that on these domains, the compactness of the ar-Neumann operator is equivalent to the compactness of all Hankel operators with smooth symbols.

Findings

Compactness of ar-Neumann operator  equivalent to Hankel operators' compactness.

Results apply specifically to smooth bounded pseudoconvex Hartogs domains in Cb2.

Provides a characterization linking two key operator properties in several complex variables.

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.