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

TL;DR

This paper constructs explicit compactly supported solutions to the Cauchy-Riemann equation in several complex variables, analyzing their $L^r$ hypoellipticity and potential generalizations to specific domains.

Contribution

It provides an explicit convolution-based method for constructing compactly supported solutions in $C^n$, extending classical results from one variable.

Findings

Constructed explicit solutions with controlled $L^p$ norms

Analyzed $L^r$ hypoellipticity of these solutions

Explored generalizations to domains with zero loci of holomorphic functions

Abstract

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.