Closed range of $\bar\partial$ on unbounded domains in $\mathbb C^n$

TL;DR

This paper introduces a new sufficient condition called weak Z(q) for the closed range property of the $ar ext{ ext}{}$ operator on unbounded domains in complex space, extending classical results to more general settings.

Contribution

It generalizes the classical Z(q) condition to weak Z(q), providing a broader criterion for closed range of $ar ext{ ext}{}$ on unbounded, non-pseudoconvex domains in $ ext{ ext}{}^n$.

Findings

Established a sufficient condition for closed range in weighted $L^2$ and Sobolev spaces.

Demonstrated the necessity of weighted spaces for closed range properties.

Provided examples illustrating the generalization beyond classical pseudoconvex domains.

Abstract

In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator $\bar\partial$ in appropriately weighted $L^2$ and $L^2$-Sobolev spaces on $(0,q)$-forms for a fixed $q$ on domains in $\mathbb{C}^n$. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical $Z(q)$ condition that we call weak $Z(q)$. We provide examples that explain the necessity of working in weighted spaces both for closed range in $L^2$ and even more critically, in $L^2$-Sobolev spaces.