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.

26 pages, Final version accepted for publication in The Journal of Geometric Analysis