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

Abstract: Let $$u: \Omega \subset \mathbb Cⁿ \to \mathbb C<sup>m$$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)² -1 \leq q &lt; \infty$ such that $\displaystyle \frac{1}{p} + \frac{1}{p&#39;} = 1$ and $\displaystyle \frac{1}{p} - \frac{1}{p&#39;} = \frac{1}{q}$. Suppose $|\overline \partial u| \leq V |u|$, where $V \in L<sup>q_{\operatorname{loc}}(\Omega)$. Then $u$ has a unique continuation property in the following sense: if $u \in W<sup>{1,p}_{\operatorname{loc}}(\Omega)$ and for some $z_0 \in \Omega$, $| u |_{L^{{p&#39;}(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 ($m=1$).