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

We obtain uniform estimates for the canonical solution to $\bar{\mathrm{\partial }}u=f$ 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.