Unique continuation of Schrödinger-type equations for $\bar{\mathrm{\partial }}$ II

In this paper, we extend our earlier unique continuation results \cite{PZ2} for the Schrödinger-type inequality $\mathrm{\mid }\bar{\mathrm{\partial }}u\mathrm{\mid }\le V\mathrm{\mid }u\mathrm{\mid }$ on a domain in ${\mathbb{C}}^n$ by removing the smoothness assumption on solutions $u=(u_1,\dots ,u_N)$. More specifically, we establish the unique continuation property for $W_{loc}^{1,1}$ solutions when the potential $V\in L_{loc}^p$, $p>2n$; and for $W_{loc}^{1,2n+\epsilon }$ solutions when $V\in L_{loc}^{2n}$ with $N=1$ or $n=2$. Although the unique continuation property fails in general if $V\in L_{loc}^p,p<2n$, we show that the property still holds for $W_{loc}^{1,1}$ solutions when $V$ is a small constant multiple of $\frac{1}{\mathrm{\mid }z\mathrm{\mid }}$.