A unique continuation property for $|\overline \partial u| \leq V |u|$

Abstract

Let $u: \Omega \subset \mathbb C^n \to \mathbb C^m$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)^2 -1 \leq q < \infty$ such that $\displaystyle \frac{1}{p} + \frac{1}{p'} = 1$ and $\displaystyle \frac{1}{p} - \frac{1}{p'} = \frac{1}{q}$. Suppose $|\overline \partial u| \leq V |u|$, where $V \in L^q_{\operatorname{loc}}(\Omega)$. Then $u$ has a unique continuation property in the following sense: if $u \in W^{1,p}_{\operatorname{loc}}(\Omega)$ and for some $z_0 \in \Omega$, $\| u \|_{L^{p'}(B(z_0,r))} $ decays faster than any powers of $r$ as $r \to 0$, then $u \equiv 0$. The same result holds for $q=\infty$ if $u$ is scalar-valued ($m=1$).