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

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.