  In one complex variable, the existence of a compactly supported solution to
the Cauchy-Riemann equation is related to the vanishing of certain integrals of
the data; trying to generalize this approach, we find an explicit construction,
via convolution, for a compactly supported solution in $\mathbb{C}^n$, which
allows to estimate the $L^p$ norm of the solution. We also investigate the
possible generalizations of this method to domains of the form $P\setminus Z$,
where $P$ is a polydisc and $Z$ is the zero locus of some holomorphic function.
