For certain annuli in $\mathbb{ C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{ \partial}$-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 $\bar{ \partial}$-equation on the non-smooth domains which are the holes of these annuli.
Categories: math.CV