Unique continuation of Schr\"odinger-type equations for $\bar\partial$ II

TL;DR

This paper extends unique continuation results for Schr"odinger-type equations in complex spaces, removing smoothness assumptions and establishing conditions under which solutions exhibit unique continuation.

Contribution

It generalizes previous results by proving unique continuation for less regular solutions and broader potential classes, including cases with singular potentials.

Findings

Unique continuation holds for $W_{loc}^{1,1}$ solutions with $V otin L_{loc}^p, p<2n$

Established unique continuation for solutions with $V otin L_{loc}^{2n}$ under higher regularity

Showed that unique continuation can still hold for certain singular potentials like multiples of $1/|z|$

Abstract

In this paper, we extend our earlier unique continuation results \cite{PZ2} for the Schr\"odinger-type inequality $ |\bar\partial u| \le V|u|$ on a domain in $\mathbb C^n$ by removing the smoothness assumption on solutions $u = (u_1, \ldots, u_N)$. 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}{|z|}$.