Sobolev regularity of the $\bar{\partial}$-equation on the Hartogs triangle

TL;DR

This paper investigates the regularity properties of the $ar{ ext{d}}$-problem on the Hartogs triangle in complex analysis, demonstrating weighted Sobolev space estimates and showing that singularities do not propagate beyond the domain's singular point.

Contribution

It provides new weighted Sobolev space regularity results for the $ar{ ext{d}}$-problem on the Hartogs triangle, highlighting the non-propagation of singularities.

Findings

Canonical solution regularity in weighted Sobolev spaces

Singularity of the Bergman projection is localized at the domain's singularity

Weighted estimates do not propagate singularities beyond (0,0)

Abstract

The regularity of the $\bar{\partial}$-problem on the domain $\{|{z_1}|<|{z_2}|<1\}$ in $\mathbb{C}^2$ is studied using $L^2$ methods. Estimates are obtained for the canonical solution in weighted $L^2$-Sobolev spaces with a weight that is singular at the point $(0,0)$. The canonical solution for $\dbar$ with weights is exact regular in the weighted Sobolev spaces away from the singularity $(0,0)$. In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.