A unique continuation property for $\lvert\overline \partial u\rvert ≤ V \lvert u\rvert$ - Shi, Ziming (MATHEMATICA SCANDINAVICA, 2025)
Let $u\colon Ω \subset \mathbb {C}^n → \mathbb {C}^m$, for $n ≥ 2$ and $m ≥ 1$. Let $1 ≤ p ≤ 2$, and $2(2n)^2 -1 ≤ q < ∞ $ such that $\frac {1}{p} + \frac {1}{p'} = 1$ and $\frac {1}{p} - \frac {1}{p'} = \frac {1}{q}$. Suppose $\lvert \overline \partial u\rvert ≤ V \lvert u\rvert $, where $V \in L^q_{loc}(Ω )$. Then $u$ has a unique continuation property in the following sense: if $u \in W^{1,p}_{loc}(Ω )$ and for some $z_0 \in Ω $, $\lVert u \rVert _{L^{p'}(B(z_0,r))} $ decays faster than any powers of $r$ as $r → 0$, then $u \equiv 0$. The same result holds for $q=∞ $ if $u$ is scalar-valued ($m=1$)