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

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.

Final version; to appear in Mathematische Annalen