A unique continuation property for $|\overline{\partial }u|\le V|u|$

Abstract

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

28 pages