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

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

2011-11-15

arxiv.org/abs/1111.3458v1

Mathematics

Complex Variables

Scientific paper

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.

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