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In one complex variable, the existence of a compactly supported solution tothe Cauchy-Riemann equation is related to the vanishing of certain integrals ofthe data; trying to generalize this approach, we find an explicit construction,via convolution, for a compactly supported solution in $\mathbb{C}^n$, whichallows to estimate the $L^p$ norm of the solution. We also investigate thepossible 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.

On $L^r$ hypoellipticity of solutions with compact support of the Cauchy-Riemann equation