Uniform estimates for the canonical solution to the $\bar\partial$-equation on product domains

Abstract

We obtain uniform estimates for the canonical solution to $\bar\partial u=f$ on the Cartesian product of bounded planar domains with $C^2$ boundaries, when $f$ is continuous up to the boundary. This generalizes Landucci's result for the bidisc toward higher dimensional product domains. In particular, it answers an open question of Kerzman for continuous datum.