Unique continuation of Schr\"odinger-type equations for ∂ II

In this paper, we extend our earlier unique continuation results {PZ2} for the Schr\"odinger-type inequality |∂ u| ≤ V|u| on a domain in Cⁿ by removing the smoothness assumption on solutions u = (u₁, …, uN). More specifically, we establish the unique continuation property for Wloc1,1 solutions when the potential V∈ Llocᵖ, $ p>2n$; and for Wloc1,2n+ε solutions when V∈ Lloc²ⁿ with $N=1$ or $n = 2$. Although the unique continuation property fails in general if V∈ Llocᵖ, p<2n, we show that the property still holds for Wloc1,1 solutions when V is a small constant multiple of 1/|z|.