  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.
