The Cauchy-Riemann equations on product domains

We establish the $L^2$ theory for the Cauchy-Riemann equations on product domains provided that the Cauchy-Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on $(p,1)$-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.