The Cauchy-Riemann equations on product domains

TL;DR

This paper develops an $L^2$ theoretical framework for the Cauchy-Riemann equations on product domains, demonstrating regularity of solutions under certain conditions and extending results to Sobolev spaces and smooth data.

Contribution

It introduces an $L^2$ theory for the Cauchy-Riemann equations on product domains with closed range assumptions, and establishes regularity results for solutions in tensor product Sobolev spaces.

Findings

Established $L^2$ theory for product domain Cauchy-Riemann equations.

Proved regularity of solutions in tensor product Sobolev spaces.

Extended regularity results to smooth data cases.

Abstract

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.