Extendability and the $\overline \partial$ Operator on the Hartogs Triangle

TL;DR

This paper demonstrates that the Hartogs triangle is a Sobolev extension domain and that the weak and strong maximal extensions of the Cauchy-Riemann operator coincide, with applications to Dolbeault cohomology with Sobolev coefficients.

Contribution

It establishes the extendability properties of the $ar{ ext{partial}}$ operator on the Hartogs triangle, showing it is a uniform and Sobolev extension domain, and compares maximal extensions.

Findings

Hartogs triangle is a uniform domain

Hartogs triangle is a Sobolev extension domain

Weak and strong maximal extensions of the Cauchy-Riemann operator agree

Abstract

In this paper it is shown that the Hartogs triangle $\mathbf T$ in $\mathbf C^2$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $\mathbf T$.