On the $L^2$-Dolbeault cohomology of annuli

TL;DR

This paper investigates the $L^2$-Dolbeault cohomology of annuli in complex spaces with non-smooth holes, establishing closed range properties for the $ar{ ext{d}}$-operator and deriving Sobolev estimates for the $ar{ ext{d}}$-equation.

Contribution

It demonstrates the closed range property of the $ar{ ext{d}}$-operator on certain non-smooth annuli and provides Sobolev space estimates for solving the $ar{ ext{d}}$-equation.

Findings

The $ar{ ext{d}}$-operator has closed range on specified non-smooth annuli.

Sobolev $W^1$ estimates are obtained for the $ar{ ext{d}}$-equation.

Applicable to holes that are products or intersections of pseudoconvex domains.

Abstract

For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli.