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

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 $\mathbf T$.