Uniform estimates of the Cauchy-Riemann equation on product domains

We observe that the continuity assumption on $f$ for the uniform estimates of the canonical solution to $\bar{\mathrm{\partial }}u=f$ 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 $\bar{\mathrm{\partial }}$ is obtained for all $p\in [1,\mathrm{\infty }]$.