We answer a question raised by Kerzman in 1971. More precisely, we show that the canonical solution of the $\overline{\mathrm{\partial }}$-equation satisfies the $L^p$ estimate on the polydisc for $p\in [1,\mathrm{\infty }]$. Moreover, the $L^p$ estimates for $p\in [1,\mathrm{\infty }]$ of $\overline{\mathrm{\partial }}$ can also be obtained on the product of bounded $C^2$ planar domains by an observation based on the method developed in [Dong et al. arXiv:2006.14484].