We observe that the continuity assumption on $f$ for the uniform estimates of the canonical solution to $\bar\partial u = f$ on products of $C^2$ bounded planar domains in \cite{ DPZ} can be reduced to the boundedness assumption. This completely answers the original question raised by Kerzman in 1971. Moreover, the $L^p$ estimates of $\bar\partial$ is obtained for all $p \in [1, \infty]$.
Categories: math.CV