On the $L^2$-Dolbeault cohomology of annuli

For certain annuli in ${\mathbb{C}}^n$, $n\ge 2$, with non-smooth holes, we show that the $\bar{\mathrm{\partial }}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\mathrm{\partial }}$-equation on the non-smooth domains which are the holes of these annuli.