Poisson equation beyond its natural domain of definition

Abstract

This note is concerned with the Poisson equation in the unit disk of the complex plane. Our setting is in the Sobolev space ${\mathcal{W}}^{1,p}(\mathbb{D})$ with exponent $1<p<\infty$. Such a setting with $p\ne 2$ is referred to as beyond the natural domain of definition. The novelty lies in the use of a singular integral called Beurling Transform.