Abstract
In this article, we extend our earlier unique continuation results of Pan and Zhang (2025,
Comm. Anal. Geom.
33, 1975–2014) for the Schrödinger-type inequality
|
∂
¯
u
|
≤
V
|
u
|
$ |\bar \partial u| \le V|u|$
StartAbsoluteValue ModifyingAbove partial differential With bar u EndAbsoluteValue less than or equals upper V StartAbsoluteValue u EndAbsoluteValue
on a domain in
C
n
$\mathbb C^n$
double struck upper C Superscript n
, by removing the smoothness assumption on solutions
u
=
(
u
1
,
…
,
u
N
)
$u = (u_1, \ldots , u_N)$
u equals left parenthesis u 1 comma ellipsis comma u Subscript upper N Baseline right parenthesis
. More specifically, we establish the unique continuation property for
W
l
o
c
1
,
1
$W_{loc}^{1,1}$
upper W Subscript l o c Superscript 1 comma 1
solutions when the potential
V
∈
L
l
o
c
p
$V\in L_{loc}^p $
upper V element of upper L Subscript l o c Superscript p
,
p
>
2
n
$ p>2n$
p greater than 2 n
, and for
W
l
o
c
1
,
2
n
+
ϵ
$W_{loc}^{1,2n+\epsilon }$
upper W Subscript l o c Superscript 1 comma 2 n plus epsilon
solutions when
V
∈
L
l
o
c
2
n
$V\in L_{loc}^{2n}$
upper V element of upper L Subscript l o c Superscript 2 n
with
N
=
1
$N=1$
upper N equals 1
or
n
=
2
$n = 2$
n equals 2
. Although the unique continuation property fails in general if
V
∈
L
l
o
c
p
,
p
<
2
n
$V\in L_{loc}^{p}, p<2n$
upper V element of upper L Subscript l o c Superscript p Baseline comma p less than 2 n
, we show that the property still holds for
W
l
o
c
1
,
1
$W_{loc}^{1,1}$
upper W Subscript l o c Superscript 1 comma 1
solutions when
V
$V $
upper V
is a small constant multiple of
1
|
z
|
$ \frac {1}{|z|}$
StartFraction 1 Over StartAbsoluteValue z EndAbsoluteValue EndFraction
. The key idea that allows us to lower the regularity assumption is a careful analysis of how regularity and flatness are transferred to the sliced functions along complex radial directions.