On Lr hypoellipticity of solutions with compact support of the Cauchy-Riemann equation

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 ℂn, which allows us to estimate the Lp norm of the solution. We also investigate the possible generalizations of this method to domains of the form P\Z, where P is a polydisc and Z is the zero locus of some holomorphic function. © 2012 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.