On weighted compactness of commutator of semi-group maximal function and fractional integrals associated to Schrödinger operators

• Published in: Revista matemática complutense Vol. 35; no. 3; pp. 871 - 893
• Main Authors: Wang, Shifen, Xue, Qingying
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-09-2022; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Commutators; Fractional calculus; Geometry; Integrals; Linear operators; Mathematics; Mathematics and Statistics; Operators (mathematics); Topology; Commutator; Compactness; Schrödinger operator; Semi-group maximal operator; Fractional integrals; Primary 42B25; Secondary 35J10
• ISSN: 1139-1138; 1988-2807
• DOI: 10.1007/s13163-021-00409-8

Let
T
∗
and
I
α
be the semi-group maximal function and fractional integrals associated to the Schrödinger operator
-
Δ
+
V
(
x
)
, respectively, with
V
satisfying an appropriate reverse Hölder inequality. In this paper, we show that the commutator of
T
∗
is a compact operator on
L
p
(
w
)
for
1
<
p
<
∞
if
b
∈
CMO
θ
(
ρ
)
(
R
n
)
and
w
∈
A
p
ρ
,
θ
(
R
n
)
. We also show that the commutator of
I
α
is a compact operator from
L
p
(
w
p
)
to
L
q
(
w
q
)
for
0
<
α
<
n
,
1
<
p
<
α
/
n
,
1
q
=
1
p
-
α
n
if
w
∈
A
(
p
,
q
)
ρ
and
b
∈
CMO
θ
(
ρ
)
(
R
n
)
. Here
CMO
θ
(
ρ
)
(
R
n
)
denotes the closure of
C
c
∞
(
R
n
)
in the
BMO
θ
(
ρ
)
(
R
n
)
(which is larger than the classical
BMO
(
R
n
)
space) topology. The space where
b
belongs and the weighs class
w
belongs are more larger than the usual
CMO
(
R
n
)
space and the Muckenhoupt
A
p
weights class, respectively.