Unique continuation for $\bar\partial$ with square-integrable potentials

TL;DR

This paper studies the unique continuation property for vector-valued functions satisfying a differential inequality involving an $L^2$ potential, establishing results for both one-dimensional and higher-dimensional cases with optimal conditions.

Contribution

It proves the strong unique continuation for $n=1$ and the weak for $n extgreater 1$, with optimal $L^2$ potential conditions.

Findings

Strong unique continuation for $n=1$

Weak unique continuation for $n extgreater 1$

Optimal $L^2$ integrability condition on potential

Abstract

In this paper, we investigate the unique continuation property for the inequality $|\bar\partial u| \le V|u|$, where $u$ is a vector-valued function from a domain in $\mathbb C^n$ to $\mathbb C^N$, and the potential $V\in L^2$. We show that the strong unique continuation property holds when $n=1$, and the weak unique continuation property holds when $n\ge 2$. In both cases, the $L^2$ integrability condition on the potential is optimal.