A remark on unique continuation for the Cauchy-Riemann operator

TL;DR

This paper establishes a unique continuation property for solutions to a differential inequality involving the Cauchy-Riemann operator and an $L^2$ potential, contributing to the understanding of complex differential inequalities.

Contribution

It provides a new unique continuation result for the Cauchy-Riemann operator with $L^2$ potentials, extending previous knowledge in complex analysis and PDEs.

Findings

Proves unique continuation for $|ar{ abla} u| \,\leq \,|V u|$ with $V \,\in L^2$

Extends classical results to less regular potentials

Offers new insights into complex differential inequalities

Abstract

In this note we obtain a unique continuation result for the differential inequality $|\bar{\partial}u|\leq|Vu|$, where $\bar{\partial}=(i\partial_y+\partial_x)/2$ denotes the Cauchy-Riemann operator and $V(x,y)$ is a function in $L^2(\mathbb{R}^2)$.