Affiliation:
1. Department of Mathematics , Jiaozuo University , 454003 , Jiaozuo , P. R. China
Abstract
Abstract
Let
ℒ
2
=
(
-
Δ
)
2
+
V
2
{\mathcal{L}_{2}=(-\Delta)^{2}+V^{2}}
be the Schrödinger-type operator on
ℝ
n
{\mathbb{R}^{n}}
(
n
≥
5
{n\geq 5}
), let
H
ℒ
2
1
(
ℝ
n
)
{H^{1}_{\mathcal{L}_{2}}(\mathbb{R}^{n})}
be the Hardy space related to
ℒ
2
{\mathcal{L}_{2}}
, and let
BMO
θ
(
ρ
)
{\mathrm{BMO}_{\theta}(\rho)}
be the BMO-type space introduced by Bongioanni, Harboure and Salinas.
In this paper, we investigate the boundedness of commutator
[
b
,
T
α
,
β
,
j
]
{[b,T_{\alpha,\beta,j}]}
, which is generated by the Riesz transform
T
α
,
β
,
j
=
V
2
α
∇
j
ℒ
2
-
β
{T_{\alpha,\beta,j}=V^{2\alpha}\nabla^{j}\mathcal{L}_{2}^{-\beta}}
,
j
=
1
,
2
,
3
{j=1,2,3}
, and
b
∈
BMO
θ
(
ρ
)
{b\in\mathrm{BMO}_{\theta}(\rho)}
. Here,
0
<
α
≤
1
-
j
4
{0<\alpha\leq 1-\frac{j}{4}}
,
j
4
<
β
≤
1
{\frac{j}{4}<\beta\leq 1}
,
β
-
α
=
j
4
{\beta-\alpha=\frac{j}{4}}
, and the nonnegative potential V
belongs to both the reverse Hölder class
RH
s
{\mathrm{RH}_{s}}
with
s
≥
n
2
{s\geq\frac{n}{2}}
and the Gaussian class associated with
(
-
Δ
)
2
{(-\Delta)^{2}}
.
The
L
p
{L^{p}}
boundedness of
[
b
,
T
α
,
β
,
j
]
{[b,T_{\alpha,\beta,j}]}
is obtained, and it is also shown that
[
b
,
T
α
,
β
,
j
]
{[b,T_{\alpha,\beta,j}]}
is bounded from
H
ℒ
2
1
(
ℝ
n
)
{H^{1}_{\mathcal{L}_{2}}(\mathbb{R}^{n})}
to weak
L
1
(
ℝ
n
)
{L^{1}(\mathbb{R}^{n})}
.
Funder
Natural Science Foundation of Henan Province
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