A unique continuation property for $\mathrm{\mid }\bar{\mathrm{\partial }}u\mathrm{\mid }\le V\mathrm{\mid }u\mathrm{\mid }$

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