# Boundedness for some Schrödinger Type Operators on Weighted Morrey Spaces

Authors: Guixia Pan; Lin Tang
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We establish the boundedness of some Schrödinger type operators on weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.


---

## Body

## 1. Introduction

In this paper, we consider the Schrödinger differential operator(1)
ℒ
=
-
Δ
+
V
(
x
)
on
ℝ
n
,
n
≥
3
,
where V
(
x
) is a nonnegative potential belonging to the reverse Hölder class B
q for q
≥
n
/
2.A nonnegative locallyL
q integrable function V
(
x
) on ℝ
n is said to belong to B
q
(
q
>
1
) if there exists C
>
0 such that the reverse Hölder inequality
(2)
(
1
|
B
|
∫
B
‍
V
q
d
x
)
1
/
q
≤
C
(
1
|
B
|
∫
B
‍
V
d
x
)
holds for every ball B in ℝ
n; see [1].Forx
∈
ℝ
n, the function ρ
(
x
) is defined by
(3)
ρ
(
x
)
∶
=
sup
r
>
0
{
r
:
1
r
n
-
2
∫
B
(
x
,
r
)
‍
V
(
y
)
d
y
≤
1
}
.
Let p
∈
[
1
,
∞
),  α
∈
(
-
∞
,
∞
) and λ
∈
[
0,1
). For f
∈
L
loc
p
(
ℝ
n
) and V
∈
B
q
(
q
>
1
), we say f
∈
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
) (weighted Morrey spaces related to the potential V) provided that
(4)
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
=
sup
B
(
x
0
,
r
)
⊂
ℝ
n
(
1
+
r
ρ
(
x
0
)
)
α
ω
(
B
(
x
0
,
2
r
)
)
-
λ
×
∫
B
(
x
0
,
r
)
‍
|
f
(
x
)
|
p
ω
(
x
)
d
x
<
∞
,
where B
=
B
(
x
0
,
r
) denotes a ball with centered at x
0 and radius r, and the weight functions ω
∈
A
p
ρ
,
∞ (see Section 2). The space L
α
,
V
,
ω
p
,
λ
(
ℝ
n
) could be viewed as an extension of weighted Lebesgue spaces (i.e., when α
=
λ
=
0, ∥
f
∥
L
0
,
V
,
ω
p
,
0
(
ℝ
n
)
=
∥
f
∥
L
ω
p
(
ℝ
n
)). In particular, when α
=
0 or V
=
0,  ω
=
1 and 0
<
λ
<
1, the space L
α
,
V
,
ω
p
,
λ
(
ℝ
n
) is the classic Morrey space L
p
,
λ
(
ℝ
n
) (see [2]). When α
=
0 or V
=
0 and 0
<
λ
<
1, L
ω
p
,
λ
(
ℝ
n
) was first introduced in [3] (see also [4]), where ω
∈
A
p
(
ℝ
n
) (Muckenhoupt weights class). It is easy to see that L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
⊂
L
ω
p
,
λ
(
ℝ
n
) for α
>
0, and L
ω
p
,
λ
(
ℝ
n
)
⊂
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
) for α
<
0. In addition, when ω
=
1, the L
α
,
V
,
ω
p
,
λ
(
ℝ
n
) has been studied in [5].From [1, 6], we know some Schrödinger type operators, such as ∇
(
-
Δ
+
V
)
-
1
∇ with V
∈
B
n, ∇
(
-
Δ
+
V
)
-
1
/
2 with V
∈
B
n, (
-
Δ
+
V
)
-
1
/
2
∇ with V
∈
B
n, (
-
Δ
+
V
)
i
γ with γ
∈
ℝ and V
∈
B
n
/
2, and ∇
2
(
-
Δ
+
V
)
-
1 with V is a nonnegative polynomial, are standard Calderón-Zygmund operators; see [7]. In particular, the kernels K of operators above all satisfy
(5)
|
K
(
x
,
y
)
|
≤
C
k
(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
k
1
|
x
-
y
|
n
for any k
∈
ℕ. Hence, in the rest of this paper, we always assume that T denotes the above operators.Recently, Bongioanni et al. [8] proved L
p
(
ℝ
n
)
(
1
<
p
<
∞
) boundedness for commutators of Riesz transforms associated with Schrödinger operator with BMO
ρ functions which include the classic BMO function, and they [9] established the weighted boundedness for Riesz transforms, fractional integrals, and Littlewood-Paley functions associated with Schrödinger operator with weight A
p
ρ
,
∞ class which includes the Muckenhoupt weight class. Very recently, the author [10, 11] established the weighted norm inequalities for some Schrödinger type operators, which include commutators of Riesz transforms, fractional integrals, and Littlewood-Paley functions with BMO
ρ functions; see also [12, 13].The aim of this paper is to study the boundedness properties of some Schrödinger type operators on the weighted Morrey spaces∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
). Our main results in this paper are formulated as follows.Theorem 1.
Supposeα
∈
(
-
∞
,
∞
) and λ
∈
(
0,1
). (i)
If1
<
p
<
∞ and ω
∈
A
p
ρ
,
∞, then
(6)
∥
T
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
,
where C is independent of f.
(ii)
Ifp
=
1 and ω
∈
A
1
ρ
,
∞, then for any t
>
0,
(7)
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
t
ω
(
{
y
∈
B
(
x
,
r
)
:
|
T
f
(
y
)
|
>
t
}
)
≤
C
∥
f
∥
L
α
,
V
,
ω
1
,
λ
(
ℝ
n
)
holds for all balls B, where C is independent of x, r, t, and f.Letb
∈
BMO
ρ (see its definition in Section 2); we define the commutator of T by
(8)
[
b
,
T
]
f
=
b
T
f
-
T
(
b
f
)
.Theorem 2.
Supposeb
∈
B
M
O
ρ, α
∈
(
-
∞
,
∞
) and λ
∈
(
0,1
). (i)
If1
<
p
<
∞ and ω
∈
A
p
ρ
,
∞, then
(9)
∥
[
b
,
T
]
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
,
where C is independent of f.
(ii)
Ifp
=
1 and ω
∈
A
1
ρ
,
∞, then, for any t
>
0,
(10)
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
ω
(
{
y
∈
B
(
x
,
r
)
:
|
[
b
,
T
]
f
(
y
)
|
>
t
}
)
≤
C
sup
B
(
x
,
r
)
⊂
ℝ
n
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
t
ln
(
2
+
|
f
(
y
)
|
t
)
ω
(
y
)
d
y
holds for all balls B
(
x
,
r
), where C is independent of x, r, t, and f.Next, we discuss the Littlewood-Paleyg function related to Schrödinger operators defined by
(11)
g
(
f
)
(
x
)
=
(
∫
0
∞
‍
|
d
d
t
e
-
t
L
(
f
)
(
x
)
|
2
t
d
t
)
1
/
2
,
and the commutator g
b of g with b
∈
BMO
(
ρ
) is defined by
(12)
g
b
(
f
)
(
x
)
=
(
∫
0
∞
‍
|
d
d
t
e
-
t
L
(
(
b
(
x
)
-
b
(
·
)
)
f
)
(
x
)
|
2
t
d
t
)
1
/
2
.Theorem 3.
SupposeV
∈
B
n
/
2, α
∈
(
-
∞
,
∞
), and λ
∈
(
0,1
). (i)
If1
<
p
<
∞ and ω
∈
A
p
ρ
,
∞, then
(13)
∥
g
(
f
)
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
,
where C is independent of f.
(ii)
Ifp
=
1 and ω
∈
A
1
ρ
,
∞, then, for any t
>
0,
(14)
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
t
ω
(
{
y
∈
B
(
x
,
r
)
:
|
g
(
f
)
(
y
)
|
>
t
}
)
≤
C
∥
f
∥
L
α
,
V
,
ω
1
,
λ
(
ℝ
n
)
holds for all balls B, where C is independent of x, r, t, and f.Theorem 4.
SupposeV
∈
B
n
/
2, b
∈
B
M
O
ρ, α
∈
(
-
∞
,
∞
), and λ
∈
(
0,1
). (i)
If1
<
p
<
∞ and ω
∈
A
p
ρ
,
∞, then
(15)
∥
g
b
(
f
)
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
,
where C is independent of f.
(ii)
Ifp
=
1 and ω
∈
A
1
ρ
,
∞, then, for any t
>
0,
(16)
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
ω
(
{
y
∈
B
(
x
,
r
)
:
|
g
b
(
f
)
(
y
)
|
>
t
}
)
≤
C
sup
B
(
x
,
r
)
⊂
ℝ
n
(
1
+
r
ρ
(
x
)
)
α
ω
(
B
(
x
,
2
r
)
)
-
λ
×
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
t
ln
(
2
+
|
f
(
y
)
|
t
)
ω
(
y
)
d
y
holds for all balls B, where C is independent of x, r, t, and f.Finally, we consider the boundedness of fractional integrals related to Schrödinger operators.Letℒ
=
-
Δ
+
V with V
∈
B
q for q
≥
n
/
2 and its associated semigroup:
(17)
T
t
f
(
x
)
=
e
-
t
ℒ
f
(
x
)
=
∫
ℝ
n
k
t
(
x
,
y
)
f
(
y
)
d
y
,
f
∈
L
2
(
ℝ
n
)
,
t
>
0
.Theℒ-fractional integral operator is defined by
(18)
ℐ
β
f
(
x
)
=
ℒ
-
β
/
2
f
(
x
)
=
∫
0
∞
‍
e
-
t
ℒ
f
(
x
)
t
β
/
2
-
1
d
t
for
0
<
β
<
n
.Theorem 5.
SupposeV
∈
B
n
/
2, α
∈
(
-
∞
,
∞
) and 0
<
β
<
n. (i)
If1
<
p
<
n
/
β,  1
/
q
=
1
/
p
-
β
/
n,  ν
=
q
/
p,  0
<
λ
<
1
/
ν, and ω
q
∈
A
1
+
q
/
p
′
ρ
,
∞, where p
′
=
p
/
(
p
-
1
), then
(19)
∥
ℐ
β
f
∥
L
α
,
V
,
ω
q
q
,
ν
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
,
where C is independent of f.
(ii)
Ifp
=
1,  q
=
n
/
(
n
-
β
),  0
<
λ
<
1, and ω
∈
A
1
ρ
,
∞, then, for any t
>
0,
(20)
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
t
ω
(
{
y
∈
B
(
x
,
r
)
:
|
ℐ
β
f
(
y
)
|
>
t
}
)
1
/
q
≤
C
∥
f
∥
L
α
,
V
,
ω
1
/
q
1
,
λ
(
ℝ
n
)
holds for all balls B
(
x
,
r
), where C is independent of x
,
r
,
t, and f.Letb
∈
BM
O
ρ; we define the commutator of ℐ
β by
(21)
[
b
,
ℐ
β
]
f
=
b
ℐ
β
f
-
ℐ
β
(
b
f
)
.Theorem 6.
Letb
∈
B
M
O,  V
∈
B
n
/
2,  α
∈
(
-
∞
,
∞
), and 0
<
β
<
n. (i)
If1
<
p
<
n
/
β,  1
/
q
=
1
/
p
-
β
/
n,  ν
=
q
/
p,  0
<
λ
<
1
/
ν, and ω
q
∈
A
1
+
q
/
p
′
ρ
,
∞, then
(22)
∥
[
b
,
ℐ
β
]
f
∥
L
α
,
V
,
ω
q
q
,
ν
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
,
where C is independent of f.
(ii)
Ifp
=
1,  q
=
n
/
(
n
-
β
),  0
<
λ
<
1, and ω
∈
A
1
ρ
,
∞, then, for any t
>
0,
(23)
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
ω
(
{
y
∈
B
(
x
,
r
)
:
|
[
b
,
ℐ
β
]
f
(
y
)
|
>
t
}
)
≤
C
sup
B
(
x
,
r
)
⊂
ℝ
n
ω
(
B
(
x
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
)
)
α
×
Φ
(
∫
B
‍
|
f
(
y
)
|
t
ln
(
2
+
|
f
(
y
)
|
t
)
Θ
(
ω
(
y
)
)
d
y
)
holds for all balls B
=
B
(
x
,
r
), where Φ
(
t
)
=
[
t
log
(
2
+
t
β
/
n
)
]
n
/
(
n
-
β
) and Θ
(
t
)
=
t
1
-
β
/
n
log
(
e
+
t
-
β
/
n
), and C is independent of x
,
r
,
t, and f.We remark that even if in theω
=
1 case, Theorems 2, 3, 4, and 6 are also new; see [5].Throughout this paper,C is a positive constant which is independent of the main parameters and not necessary the same at each occurrence.


## 2. Some Notation and Basic Results

We first recall some notation. GivenB
=
B
(
x
,
r
) and λ
>
0, we will write λ
B for the λ-dilate ball, which is the ball with the same center x and with radius λ
r. Given a Lebesgue measurable set E and a weight ω, |
E
| will denote the Lebesgue measure of E and ω
(
E
)
=
∫
E
‍
ω
d
x. ∥
f
∥
L
p
(
ω
) will denote (
∫
ℝ
n
‍
|
f
(
y
)
|
p
ω
(
y
)
d
y
)
1
/
p for 0
<
p
<
∞.Lemma 7 (see [1]).
There existsl
0
>
0 and C
0
>
1 such that
(24)
1
C
0
(
1
+
|
x
-
y
|
ρ
(
x
)
)
-
l
0
≤
ρ
(
y
)
ρ
(
x
)
≤
C
0
(
1
+
|
x
-
y
|
ρ
(
x
)
)
l
0
/
(
l
0
+
1
)
.
In particular, ρ
(
x
)
~
ρ
(
y
) if |
x
-
y
|
<
C
ρ
(
x
).In this paper, we writeΨ
θ
(
B
)
=
(
1
+
r
/
ρ
(
x
0
)
)
θ, where θ
>
0,  x
0 and r denotes the center and radius of B,  respectively.A weight will always mean a nonnegative function which is locally integrable. As in [9], we say that a weight ω belongs to the class A
p
ρ
,
θ for 1
<
p
<
∞, if there is a constant C such that for all ball B
=
B
(
x
,
r
)
(25)
(
1
Ψ
θ
(
B
)
|
B
|
∫
B
‍
ω
(
y
)
d
y
)
×
(
1
Ψ
θ
(
B
)
|
B
|
∫
B
‍
ω
-
1
/
(
p
-
1
)
(
y
)
d
y
)
p
-
1
≤
C
.
We also say that a nonnegative function ω satisfies the A
1
ρ
,
θ condition if there exists a constant C for all balls B
(26)
M
V
θ
(
ω
)
(
x
)
≤
C
ω
(
x
)
,
a
.
e
.
x
∈
ℝ
n
,
where
(27)
M
V
θ
f
(
x
)
=
sup
x
∈
B
1
Ψ
θ
(
B
)
|
B
|
∫
B
‍
|
f
(
y
)
|
d
y
.
Since Ψ
θ
(
B
)
≥
1, obviously, A
p
⊂
A
p
ρ
,
θ for 1
≤
p
<
∞, where A
p denote the classical Muckenhoupt weights; see [7]. We will see that A
p
⊂
A
p
ρ
,
θ for 1
≤
p
<
∞ in some cases. In fact, let θ
>
0 and 0
≤
γ
≤
θ; it is easy to check that ω
(
x
)
=
(
1
+
|
x
|
)
-
(
n
+
γ
)
∉
A
∞
=
⋃
p
≥
1
‍
A
p and ω
(
x
)
d
x is not a doubling measure, but ω
(
x
)
=
(
1
+
|
x
|
)
-
(
n
+
γ
)
∈
A
1
ρ
,
θ provided that V
=
1 and Ψ
θ
(
B
(
x
0
,
r
)
)
=
(
1
+
r
)
θ.For convenience, we always assume thatΨ
(
B
) denotes Ψ
θ
(
B
), A
p
ρ
,
∞
=
⋃
θ
>
0
‍
A
p
ρ
,
θ, and A
∞
ρ
,
∞
=
⋃
p
≥
1
‍
A
p
ρ
,
∞.Lemma 8 (see [10]).
Let0
<
θ
<
∞; then (i)
if1
≤
p
1
<
p
2
<
∞, then A
p
1
ρ
,
θ
⊂
A
p
2
ρ
,
θ.
(ii)
ω
∈
A
p
ρ
,
θ if and only if ω
-
1
/
(
p
-
1
)
∈
A
p
′
ρ
,
θ, where 1
/
p
+
1
/
p
′
=
1.
(iii)
Ifω
∈
A
p
ρ
,
θ for 1
≤
p
<
∞, then there exists a constant such that for any λ
>
1
(28)
ω
(
λ
B
(
x
0
,
r
)
)
≤
C
(
1
+
λ
r
ρ
(
x
0
)
)
(
l
0
+
1
)
θ
ω
(
B
(
x
0
,
r
)
)
.Lemma 9 (see [12]).
Let0
<
θ
<
∞, 1
≤
p
<
∞. If ω
∈
A
p
ρ
,
θ, then there exists positive constants δ
,
η, and C such that
(29)
(
1
|
B
|
∫
B
‍
ω
(
y
)
1
+
δ
d
y
)
1
/
(
1
+
δ
)
≤
C
1
|
B
|
∫
B
‍
ω
(
y
)
d
y
(
1
+
r
ρ
(
x
0
)
)
η
for all ball B
(
x
0
,
r
).As a consequence of Lemma9, we have the following result.Corollary 10 (see [12]).
Let0
<
θ
<
∞, 1
≤
p
<
∞. If ω
∈
A
p
ρ
,
θ, then there exist positive constants q
>
1, η and C such that
(30)
ω
(
E
)
ω
(
B
)
≤
C
(
|
E
|
|
B
|
)
1
/
q
(
1
+
r
ρ
(
x
0
)
)
η
for any measurable subset E of a ball B
(
x
0
,
r
).Bongioanni et al. [8] introduce a new space BMO
θ
(
ρ
) defined by
(31)
∥
f
∥
BM
O
θ
(
ρ
)
=
sup
B
⊂
ℝ
n
1
Ψ
θ
(
B
)
|
B
|
∫
B
‍
|
f
(
x
)
-
f
B
|
d
x
<
∞
,
where f
B
=
(
1
/
|
B
|
)
∫
B
‍
f
(
y
)
d
y and Ψ
θ
(
B
)
=
(
1
+
r
/
ρ
(
x
0
)
)
θ and θ
>
0.In particular, Bongioanni et al. [8] proved the following result for BMO
θ
(
ρ
).Proposition 11.
Letθ
>
0 and 1
≤
s
<
∞. If b
∈
B
M
O
θ
(
ρ
), then
(32)
(
1
|
B
|
∫
B
‍
|
b
-
b
B
|
s
)
1
/
s
≤
c
C
0
θ
s
∥
b
∥
B
M
O
θ
(
ρ
)
(
1
+
r
ρ
(
x
0
)
)
θ
′
,
for all B
=
B
(
x
0
,
r
), with x
∈
ℝ
n and r
>
0, where θ
′
=
(
l
0
+
1
)
θ and C
0 is defined in Lemma 7 and c is a constant depending only on n.Obviously, the classicalBMO is properly contained in BM
O
θ
(
ρ
); for more examples, see [8]. For convenience, we let BMO
ρ
=
⋃
θ
>
0
‍
BM
O
θ
(
ρ
).Tang [10] gave the following result, which is equivalent to Proposition 11.Proposition 12.
Suppose thatf
∈
B
M
O
θ
(
ρ
). There exist positive constants γ and C such that for any ball B
=
B
(
x
0
,
r
)
(33)
1
|
B
|
∫
B
‍
exp
{
γ
∥
f
∥
B
M
O
θ
(
ρ
)
Ψ
θ
′
(
B
)
|
f
(
x
)
-
f
B
|
}
d
x
≤
C
.Applying Corollary10 and Proposition 11, we can obtain the following result.Proposition 13.
Iff
∈
B
M
O
θ
(
ρ
) and ω
∈
A
p
ρ
,
θ  (
p
>
1
), then there exist positive constants c
1
,
c
2, and η such that for every ball B
=
B
(
x
,
r
) and every λ
>
0, one has
(34)
ω
(
{
x
∈
B
:
|
f
(
x
)
-
f
B
|
>
λ
}
)
≤
c
1
ω
(
B
)
exp
{
-
c
2
λ
∥
f
∥
B
M
O
θ
(
ρ
)
Ψ
θ
′
(
B
)
}
(
1
+
r
ρ
(
x
)
)
η
,
where f
B
=
(
1
/
|
B
|
)
∫
B
‍
f
(
y
)
d
y, Ψ
θ
′
(
B
)
=
(
1
+
r
/
ρ
(
x
0
)
)
θ
′ and θ
′
=
(
l
0
+
1
)
θ.Proof.
We adapt the same argument of pages 145-146 in [7]. We first assume ∥
f
∥
BMO
(
ρ
)
Ψ
θ
′
(
B
)
=
1. We apply Chebysheff’s inequality and Proposition 11; we obtain
(35)
|
{
x
∈
B
:
|
f
(
x
)
-
f
B
|
>
λ
}
|
≤
(
c
C
0
θ
s
)
s
λ
-
s
|
B
|
for 0
<
λ
<
∞, 1
≤
s
<
∞. From this and by Corollary 10, there exist constants q
>
1 and η such that
(36)
ω
(
{
x
∈
B
:
|
f
(
x
)
-
f
B
|
>
λ
}
)
≤
(
c
C
0
θ
s
)
s
/
q
λ
-
s
/
q
ω
(
B
)
(
1
+
r
ρ
(
x
)
)
η
,
for 0
<
λ
<
∞, 1
≤
s
<
∞.
Ifλ
≥
2
c
C
0
θ
/
q, we take s
=
λ
/
(
2
c
C
0
θ
/
q
)
≥
1. Then
(37)
ω
(
{
x
∈
B
:
|
f
(
x
)
-
f
B
|
>
λ
}
)
≤
(
1
2
)
s
ω
(
B
)
(
1
+
r
ρ
(
x
)
)
η
=
e
-
c
1
λ
ω
(
B
)
(
1
+
r
ρ
(
x
)
)
η
,
where c
1
=
(
2
c
C
0
θ
/
q
)
-
1
ln
2. However, if λ
≤
2
c
C
0
θ
/
q, then e
-
c
1
λ
≥
e
-
c
1
2
(
c
C
0
θ
/
q
)
=
1
/
2, and
(38)
ω
(
{
x
∈
B
:
|
f
(
x
)
-
f
B
|
>
λ
}
)
≤
2
e
-
c
1
λ
ω
(
B
)
(
1
+
r
ρ
(
x
)
)
η
in that range of λ. Altogether then, if we drop the normalization on f by replacing f by f
/
(
∥
f
∥
BM
O
θ
(
ρ
)
Ψ
θ
′
(
B
)
), we can obtain the conclusion by taking c
1
=
2 and c
2
=
(
2
c
C
0
θ
/
q
)
-
1
ln
2.From Proposition13, it is easy to see the following.Corollary 14.
Iff
∈
B
M
O
ρ and ω
∈
A
∞
ρ
,
∞, then there exist positive constants C and η such that for every ball B
=
B
(
x
,
r
), one has
(39)
1
ω
(
B
)
∫
B
‍
|
f
(
x
)
-
f
B
|
ω
(
x
)
d
x
≤
(
1
+
r
ρ
(
x
)
)
η
∥
f
∥
B
M
O
ρ
p
,
where f
B
=
(
1
/
|
B
|
)
∫
B
‍
f
(
y
)
d
y.


## 3. Proof of Theorems1–4

Proof of Theorem1.
Without loss of generality, we may assume thatα
<
0 and ω
∈
A
p
ρ
,
θ. Pick any ball B
=
B
(
x
0
,
r
) and write
(40)
f
(
x
)
=
f
0
(
x
)
+
∑
i
=
1
∞
‍
f
i
(
x
)
,
where f
0
=
χ
B
(
x
0
,
2
r
)
f
,
f
i
=
χ
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
)
f for i
≥
1. Hence, we have
(41)
(
∫
B
(
x
0
,
r
)
‍
|
T
f
(
x
)
|
p
ω
(
x
)
d
x
)
1
/
p
≤
(
∫
B
(
x
0
,
r
)
‍
|
T
f
0
(
x
)
|
p
d
x
)
1
/
p
a
+
∑
i
=
1
∞
‍
(
∫
B
(
x
0
,
r
)
‍
|
T
f
i
(
x
)
|
p
ω
(
x
)
d
x
)
1
/
p
.
By the L
ω
p boundedness of T   (see [10]), we obtain
(42)
∫
B
(
x
0
,
r
)
‍
|
T
f
0
(
x
)
|
p
ω
(
x
)
d
x
≤
C
(
1
+
r
ρ
(
x
0
)
)
-
α
×
ω
(
B
(
x
0
,
2
r
)
)
λ
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
.
By Lemma 8 and Corollary 10, as well as (5), there exist some positive constants q
>
1 and η such that
(43)
∫
B
(
x
0
,
r
)
‍
|
T
f
i
(
x
)
|
p
ω
(
x
)
d
x
≤
C
∫
B
(
x
0
,
r
)
‍
(
∫
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
)
‍
|
K
(
x
,
y
)
f
(
y
)
|
d
y
)
p
×
ω
(
x
)
d
x
≤
C
k
ω
(
B
(
x
0
,
2
i
+
1
r
)
)
λ
-
1
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
×
∫
B
(
x
0
,
r
)
‍
(
1
+
(
2
i
r
/
p
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
p
θ
(
1
+
(
2
i
r
/
ρ
(
x
)
)
)
k
p
d
x
≤
C
k
ω
(
B
(
x
0
,
2
i
+
1
r
)
)
λ
-
1
ω
(
B
(
x
0
,
r
)
)
×
(
1
+
(
2
i
r
/
p
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
p
θ
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
k
p
/
(
l
0
+
1
)
×
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
≤
C
k
(
ω
(
B
(
x
0
,
2
r
)
)
ω
(
B
(
x
0
,
2
i
+
1
r
)
)
)
1
-
λ
ω
(
B
(
x
0
,
2
r
)
)
λ
×
(
1
+
(
2
i
r
/
p
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
p
θ
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
k
p
/
(
l
0
+
1
)
×
∥
f
∥
L
α
,
V
p
,
λ
(
ℝ
n
)
p
≤
C
k
2
-
i
n
(
1
-
λ
)
/
q
ω
(
B
(
x
0
,
2
r
)
)
λ
×
(
1
+
(
2
i
r
/
p
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
p
θ
+
η
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
k
p
/
(
l
0
+
1
)
×
∥
f
∥
L
α
,
V
p
,
λ
(
ℝ
n
)
p
.
From (41), (42), and (43) with k
=
2
(
[
-
α
+
(
l
0
+
n
+
4
)
p
θ
+
η
]
+
1
)
(
l
0
+
1
), we obtain
(44)
∥
T
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
.
As for the case p
=
1, the proof can be given by replacing (42) with the corresponding weak estimate.Proof of Theorem2.
Without loss of generality, we may assume thatα
<
0
,
b
∈
BM
O
θ
(
ρ
). Pick any ball B
=
B
(
x
0
,
r
), as in the proof of Theorem 1 we write
(45)
f
(
x
)
=
f
0
(
x
)
+
∑
i
=
1
∞
‍
f
i
(
x
)
.
Set ω
∈
A
p
ρ
,
∞ for p
>
1. By the L
ω
p boundedness of [
b
,
T
]  (see [10]), we get
(46)
∫
B
(
x
0
,
r
)
‍
|
[
b
,
T
]
f
0
(
x
)
|
p
ω
(
x
)
d
x
≤
C
(
1
+
r
ρ
(
x
)
)
-
α
ω
(
B
(
x
0
,
2
r
)
)
λ
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
.
Set
(47)
b
r
=
1
|
B
(
x
0
,
r
)
|
∫
B
(
x
0
,
r
)
‍
b
(
x
)
d
x
.
When i
≥
1, by Lemmas 7 and 9 and Corollaries 10 and 14, there exist some positive constants q
>
1 and η such that
(48)
∫
B
(
x
0
,
r
)
‍
|
[
b
,
T
]
f
i
(
x
)
|
p
ω
(
x
)
d
x
≤
C
k
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
k
p
/
(
l
0
+
1
)
2
-
i
n
p
×
{
∫
B
(
x
0
,
r
)
‍
|
b
(
x
)
-
b
r
|
p
ω
(
x
)
d
x
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
d
y
aaa
+
ω
(
B
(
x
0
,
r
)
)
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
b
(
y
)
-
b
r
|
|
f
(
y
)
|
d
y
}
≤
C
k
i
p
ω
(
B
(
x
0
,
2
i
r
)
)
1
-
λ
ω
(
B
(
x
0
,
2
r
)
)
λ
×
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
p
θ
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
k
p
/
(
l
0
+
1
)
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
≤
C
k
i
(
ω
(
B
(
x
0
,
2
r
)
)
ω
(
B
(
x
0
,
2
i
+
1
r
)
)
)
1
-
λ
ω
(
B
(
x
0
,
2
r
)
)
λ
×
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
p
θ
+
η
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
k
p
/
(
l
0
+
1
)
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
≤
C
k
i
2
-
i
n
(
1
-
λ
)
/
q
ω
(
B
(
x
0
,
2
r
)
)
λ
×
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
p
θ
+
η
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
k
p
/
(
l
0
+
1
)
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
p
.
If we take k
=
2
(
[
-
α
+
(
l
0
+
n
+
4
)
p
θ
+
η
]
+
1
)
(
l
0
+
1
) in (48), then we obtain
(49)
∥
[
b
,
T
]
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
ℝ
n
)
.
It remains to consider the case p
=
1. Set ω
∈
A
1
ρ
,
θ. From [10], we know that for any t
>
0
(50)
ω
(
{
y
∈
ℝ
n
:
|
[
b
,
T
]
f
(
y
)
|
>
t
}
)
≤
C
∫
ℝ
n
‍
|
f
(
x
)
|
t
ln
(
2
+
|
f
(
x
)
|
t
)
ω
(
x
)
d
x
.
From this, we have
(51)
ω
(
B
(
x
0
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
α
×
ω
(
{
y
∈
B
(
x
0
,
r
)
:
|
[
b
,
T
]
f
0
(
y
)
|
>
t
}
)
≤
C
(
1
+
r
ρ
(
x
0
)
)
α
ω
(
B
(
x
0
,
2
r
)
)
×
∫
B
(
x
0
,
2
r
)
‍
|
f
(
x
)
|
t
ln
(
2
+
|
f
(
x
)
|
t
)
ω
(
x
)
d
x
.
Set
(52)
b
2
i
+
1
r
=
1
|
B
(
x
0
,
2
i
+
1
r
)
|
∫
B
(
x
0
,
2
i
+
1
r
)
‍
b
(
x
)
d
x
.
When i
≥
1, by Lemma 7, Corollary 10, and Proposition 12, note that ω
∈
A
1
ρ
,
θ, then there exist some positive constants q
>
1 and η such that
(53)
ω
(
B
(
x
0
,
2
r
)
)
-
λ
t
-
1
(
1
+
r
ρ
(
x
0
)
)
α
×
∫
B
(
x
0
,
r
)
‍
|
[
b
,
T
]
f
i
(
y
)
ω
(
y
)
|
d
y
≤
C
k
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
α
+
(
l
0
+
n
+
4
)
p
θ
+
η
(
1
+
(
r
/
ρ
(
x
0
)
)
)
k
/
(
l
0
+
1
)
×
(
2
i
r
)
-
n
ω
(
B
(
x
0
,
2
r
)
)
-
λ
t
-
1
(
1
+
r
ρ
(
x
0
)
)
α
×
{
∫
B
(
x
0
,
r
)
‍
|
b
(
y
)
-
b
2
i
+
1
r
|
ω
(
x
)
d
x
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
d
y
+
ω
(
B
(
x
0
,
r
)
)
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
b
(
y
)
-
b
2
i
+
1
r
|
|
f
(
y
)
|
d
y
}
≤
C
ω
(
B
(
x
0
,
2
r
)
)
-
λ
ω
(
B
)
t
-
1
(
1
+
r
ρ
(
x
0
)
)
-
3
k
0
×
(
∫
B
i
‍
|
f
(
y
)
|
ω
(
y
)
d
y
∥
b
-
b
2
i
+
1
r
∥
exp
L
,
B
i
∥
f
∥
L
log
L
,
B
i
+
i
ω
(
B
i
)
-
1
×
∥
b
∥
BM
O
θ
(
ρ
)
∫
B
i
‍
|
f
(
y
)
|
ω
(
y
)
d
y
)
≤
C
i
ω
(
B
(
x
0
,
2
r
)
)
-
λ
ω
(
B
)
t
-
1
(
1
+
r
ρ
(
x
0
)
)
-
3
k
0
×
inf
γ
>
0
{
γ
+
γ
|
B
i
|
∫
B
i
‍
|
f
(
y
)
|
γ
log
(
2
+
|
f
(
y
)
|
γ
)
d
y
}
+
C
i
ω
(
B
(
x
0
,
2
r
)
)
-
λ
ω
(
B
)
ω
(
B
i
)
-
1
t
-
1
(
1
+
r
ρ
(
x
0
)
)
-
3
k
0
×
∫
B
i
‍
|
f
(
y
)
|
ω
(
y
)
d
y
≤
C
i
(
ω
(
2
B
)
ω
(
B
i
)
)
1
-
λ
ω
(
2
B
i
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
-
k
0
×
∫
B
i
‍
|
f
(
y
)
|
t
log
(
2
+
|
f
(
y
)
|
t
)
ω
(
y
)
d
y
≤
C
i
2
-
i
n
(
1
-
λ
)
/
q
ω
(
2
B
i
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
α
×
∫
B
i
‍
|
f
(
y
)
|
t
log
(
2
+
|
f
(
y
)
|
t
)
ω
(
y
)
d
y
,
by taking k
=
(
2
[
-
α
+
(
l
0
+
n
+
4
)
p
θ
+
η
]
+
1
)
(
l
0
+
1
) and k
=
4
k
0, where in the sixth inequality, we used the following facts (see [2]);
(54)
∥
f
∥
L
log
L
,
B
,
ω
=
inf
{
γ
>
0
:
1
|
B
|
∫
B
‍
|
f
(
y
)
|
γ
a
a
a
a
×
log
(
2
+
|
f
(
y
)
|
γ
)
d
y
≤
10
}
,
∥
f
∥
exp
L
,
B
=
inf
{
γ
>
0
:
1
|
B
|
∫
B
‍
exp
(
|
f
(
y
)
|
γ
)
d
y
≤
10
}
;
the generalized Hölder inequality
(55)
1
|
B
|
∫
B
‍
|
f
(
y
)
h
(
y
)
|
d
y
≤
C
∥
f
∥
L
log
L
,
B
∥
h
∥
exp
L
,
B
,
∥
f
∥
L
log
L
,
B
≤
inf
λ
>
0
{
γ
+
γ
|
B
|
∫
B
‍
|
f
(
y
)
|
γ
log
(
2
+
|
f
(
y
)
|
γ
)
d
y
}
≤
2
∥
f
∥
L
log
L
,
B
.
Combing (51) and (53), we obtain that
(56)
ω
(
B
(
x
0
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
α
×
ω
(
{
y
∈
B
(
x
,
r
)
:
|
[
b
,
T
]
f
(
y
)
|
>
t
}
)
≤
C
sup
B
(
x
,
r
)
⊂
ℝ
n
(
1
+
r
ρ
(
x
0
)
)
α
ω
(
B
(
x
0
,
2
r
)
)
-
λ
×
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
t
ln
(
2
+
|
f
(
y
)
|
t
)
ω
(
y
)
d
y
holds for all balls B, where C is independent of x
,
r
,
t, and f.
Thus, Theorem2 is proved.Finally, we give some sketch proof of Theorems3 and 4.Proof of Theorems3 and 4.
Let us denoteB
=
L
2
(
ℝ
+
,
t
d
t
) the set of measurable functions a
:
ℝ
+
→
C with norm |
a
|
B
=
(
∫
ℝ
+
‍
|
a
(
y
,
t
)
|
2
t
d
t
)
1
/
2
<
∞, ℳ
(
ℝ
n
) the set of measurable functions defined on ℝ
n valued in ℂ, and ℳ
(
ℝ
n
,
B
) the set of Bochner-measurable functions h
:
ℝ
n
→
B. The space L
p
(
ℝ
n
,
B
)
(
ω
) is the set of h
∈
ℳ
(
ℝ
n
,
B
) with finite norm
(57)
∥
h
∥
L
p
(
ℝ
n
,
B
)
(
ω
)
=
(
∫
ℝ
n
‍
|
h
(
x
)
|
B
p
ω
(
x
)
d
x
)
1
/
p
.
We simply name the space as L
p
(
ℝ
n
,
B
) when ω
=
1.
Thus, we can redefine theg as follows:
(58)
g
(
f
)
(
x
)
=
{
g
t
f
(
x
)
≔
Q
t
f
(
x
)
}
t
∈
ℝ
+
,
Q
t
f
(
x
)
:
=
e
-
t
L
(
f
)
(
x
)
,
which has associated kernel
(59)
K
(
x
,
y
)
=
{
Q
t
(
x
,
y
)
}
t
∈
ℝ
+
.
It is easy to see that
(60)
|
K
(
x
,
y
)
|
B
≤
C
k
(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
k
1
|
x
-
y
|
n
for any k
∈
N.
Thus, we can adapt the same argument of Theorems1 and 2 to prove Theorems 3 and 4; we omit the details here.


## 4. Proof of Theorems5 and 6

We first need the following lemma.Lemma 15 (see [5]).
Letk
t
(
x
,
y
) be as in (17). For any N
∈
ℕ, there exists a C
N such that
(61)
∫
0
∞
‍
t
-
(
n
-
β
)
/
2
-
1
k
t
(
x
,
y
)
d
t
≤
C
N
(
1
+
(
|
x
0
-
y
|
/
ρ
(
x
0
)
)
)
N
1
|
x
0
-
y
|
n
-
β
for all x
∈
B
(
x
0
,
r
) and y
∈
ℝ
n
∖
B
(
x
0
,
2
r
).Proof of Theorem5.
Without loss of generality, we may assume thatα
<
0 and ω
q
∈
A
1
+
q
/
p
′
ρ
,
θ
(
p
>
1
). Pick any ball B
=
B
(
x
0
,
r
), as the proof of Theorem 1, we write
(62)
f
(
x
)
=
f
0
(
x
)
+
∑
i
=
1
∞
‍
f
i
(
x
)
.
Hence, we have
(63)
(
∫
B
(
x
0
,
r
)
‍
|
ℐ
β
f
(
x
)
ω
(
x
)
|
q
d
x
)
1
/
q
≤
(
∫
B
(
x
0
,
r
)
‍
|
ℐ
β
f
0
(
x
)
ω
(
x
)
|
q
d
x
)
1
/
q
+
∑
i
=
1
∞
‍
(
∫
B
(
x
0
,
r
)
‍
|
ℐ
β
f
i
(
x
)
ω
(
x
)
|
q
d
x
)
1
/
q
.
Let ν
=
q
/
p. By the L
ω
p
p
→
L
ω
q
q boundedness of ℐ
β (see [10]), we get
(64)
∫
B
(
x
0
,
r
)
‍
|
ℐ
β
f
0
(
x
)
ω
(
x
)
|
q
d
x
≤
C
(
1
+
r
ρ
(
x
0
)
)
α
ω
(
2
B
)
ν
λ
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
q
.
From (61), by Corollary 10, then there exists a constant q
0
>
1 such that
(65)
∫
B
(
x
0
,
r
)
|
ℐ
β
f
i
(
x
)
ω
(
x
)
|
q
d
x
‍
≤
C
∫
B
(
x
0
,
r
)
‍
(
∫
0
∞
∫
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
)
|
f
(
y
)
k
t
(
x
,
y
)
×
f
(
y
)
|
d
y
t
β
/
2
-
1
d
t
∫
0
∞
∫
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
)
|
k
t
(
x
,
y
)
)
q
×
ω
(
x
)
q
d
x
≤
C
N
∫
B
(
x
0
,
r
)
‍
(
2
i
r
)
-
n
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
N
q
(
∫
B
(
x
0
2
i
+
2
r
)
‍
|
f
(
y
)
|
p
d
y
)
q
/
p
×
ω
(
x
)
q
d
x
≤
C
N
(
ω
(
B
(
x
0
,
2
r
)
)
ω
(
B
(
x
0
,
2
i
+
1
r
)
)
)
1
-
λ
ν
ω
(
B
(
x
0
,
2
r
)
)
-
λ
ν
×
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
q
θ
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
N
q
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
q
≤
C
N
2
i
n
(
λ
ν
-
1
)
/
q
0
ω
(
B
(
x
0
,
2
r
)
)
-
λ
ν
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
q
,
where N
=
2
(
[
-
α
+
(
l
0
+
n
+
4
)
q
θ
]
+
1
).
Note thatν
λ
<
1. So
(66)
∥
ℐ
β
f
∥
L
α
,
V
,
ω
q
q
,
ν
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
.
As for the casep
=
1, the proof is similar.Proof of Theorem6.
Without loss of generality, we may assume thatα
<
0 and b
∈
BM
O
θ
(
ρ
). Pick any ball B
=
B
(
x
0
,
r
), as in the proof of Theorem 1, we write
(67)
f
(
x
)
=
f
0
(
x
)
+
∑
i
=
1
∞
‍
f
i
(
x
)
.
For p
>
1. Set ω
∈
A
p
∞
,
θ. Let ν
=
q
/
p. By the L
ω
p
p
→
L
ω
q
q boundedness of [
b
,
I
β
] (see [10]), we get
(68)
∫
B
(
x
0
,
r
)
‍
|
[
b
,
ℐ
β
]
f
0
(
x
)
ω
(
x
)
|
q
d
x
≤
C
(
1
+
r
ρ
(
x
0
)
)
-
α
ω
(
B
(
x
0
,
2
r
)
)
ν
λ
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
q
.
Set
(69)
b
r
=
1
|
B
(
x
0
,
r
)
|
∫
B
(
x
0
,
r
)
‍
b
(
x
)
d
x
.
When i
≥
1, by (61) and Corollary 10, then there exists some positive constant q
0
>
1 such that
(70)
∫
B
(
x
0
,
r
)
‍
|
[
b
,
ℐ
β
]
f
i
(
x
)
ω
(
x
)
|
q
d
x
≤
C
N
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
N
(
2
i
r
)
-
(
n
-
β
)
×
{
(
∫
B
(
x
0
,
r
)
‍
|
b
(
x
)
-
b
r
|
q
ω
q
(
x
)
d
x
)
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
d
y
hhhhh
+
ω
(
B
(
x
0
,
r
)
)
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
b
(
y
)
-
b
r
|
|
f
(
y
)
|
d
y
}
≤
C
N
i
∥
b
∥
BM
O
ρ
(
ω
(
B
(
x
0
,
2
r
)
)
ω
(
B
(
x
0
,
2
i
+
1
r
)
)
)
1
-
λ
ν
ω
(
B
(
x
0
,
2
r
)
)
-
λ
ν
×
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
-
α
+
(
l
0
+
n
+
4
)
q
θ
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
N
q
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
q
≤
C
N
2
i
n
(
λ
ν
-
1
)
/
q
0
ω
(
B
(
x
0
,
2
r
)
)
-
λ
ν
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
q
,
by taking N
=
2
(
[
-
α
+
(
l
0
+
n
+
4
)
q
θ
]
+
1
).
Then,(71)
∥
[
b
,
ℐ
β
]
f
∥
L
α
,
V
,
ω
q
q
,
ν
λ
(
ℝ
n
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
ℝ
n
)
.
It remains to consider the case p
=
1 and q
=
n
/
(
n
-
β
). Set ω
∈
A
1
ρ
,
θ. From [10], we know
(72)
ω
(
{
y
∈
ℝ
n
:
|
[
b
,
ℐ
β
]
f
(
y
)
|
>
t
}
)
≤
C
Φ
(
∫
ℝ
n
‍
|
f
(
x
)
|
t
ln
(
2
+
|
f
(
x
)
|
t
)
Θ
(
ω
(
x
)
)
d
x
)
,
where Φ
(
t
)
=
[
t
log
(
2
+
t
β
/
n
)
]
n
/
(
n
-
β
) and Θ
(
t
)
=
t
1
-
β
/
n
log
(
e
+
t
-
β
/
n
).
From this, we have(73)
ω
(
B
(
x
0
,
2
r
)
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
α
×
ω
(
{
y
∈
B
(
x
0
,
r
)
:
|
[
b
,
ℐ
β
]
f
0
(
y
)
|
>
t
}
)
≤
C
sup
B
(
x
0
,
r
)
⊂
ℝ
n
(
1
+
r
ρ
(
x
0
)
)
α
ω
(
B
(
x
0
,
2
r
)
)
-
λ
×
Φ
(
∫
B
(
x
0
,
2
r
)
‍
|
f
(
x
)
|
t
ln
(
2
+
|
f
(
x
)
|
t
)
Θ
(
ω
(
x
)
)
d
x
)
.
Set(74)
b
2
i
+
1
r
=
1
|
B
(
x
0
,
2
i
+
1
r
)
|
∫
B
(
x
0
,
2
i
+
1
r
)
‍
b
(
x
)
d
x
.
When i
≥
1, by (61) and Corollary 10 and Proposition 12, note that ω
∈
A
1
ρ
,
θ, then there exist some positive constants q
0
>
1 and η such that
(75)
ω
(
B
(
x
0
,
2
r
)
)
-
λ
t
-
q
(
1
+
r
ρ
(
x
0
)
)
α
×
∫
B
(
x
0
,
r
)
‍
|
[
b
,
ℐ
β
]
f
i
(
x
)
|
q
ω
(
x
)
d
x
≤
C
(
1
+
(
2
i
r
/
ρ
(
x
0
)
)
)
N
q
(
2
i
r
)
-
n
ω
(
B
(
x
0
,
2
r
)
)
-
λ
×
(
1
+
r
ρ
(
x
0
)
)
α
t
-
q
×
{
∫
B
(
x
0
,
r
)
‍
|
b
(
x
)
-
b
2
i
+
1
r
|
q
ω
(
x
)
d
x
×
(
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
d
y
)
q
+
ω
(
B
(
x
0
,
r
)
)
(
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
b
(
y
)
-
b
2
i
+
1
r
|
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
×
|
f
(
y
)
|
d
y
∫
B
(
x
0
,
2
i
+
1
r
)
‍
)
q
∫
B
(
x
0
,
2
i
+
1
r
)
‍
}
≤
C
(
1
+
2
i
r
ρ
(
x
0
)
)
-
3
N
0
(
2
i
r
)
-
n
ω
(
B
(
x
0
,
2
r
)
)
1
-
λ
t
-
q
×
(
|
B
i
|
∥
(
b
-
b
2
i
+
1
r
)
Φ
θ
′
∥
exp
L
,
B
i
∥
f
∥
L
log
L
,
B
i
+
i
∫
B
i
‍
|
f
(
y
)
|
d
y
∥
(
b
-
b
2
i
+
1
r
)
Φ
θ
′
∥
exp
L
,
B
i
)
q
≤
C
i
q
(
1
+
2
i
r
ρ
(
x
0
)
)
-
3
N
0
|
B
i
|
-
1
ω
(
B
(
x
0
,
2
r
)
)
1
-
λ
×
(
inf
γ
>
0
{
γ
+
γ
|
B
i
|
∫
B
i
‍
|
f
(
y
)
|
γ
log
(
2
+
|
f
(
y
)
|
γ
)
d
y
}
|
B
i
|
t
-
1
inf
γ
>
0
{
γ
+

γ
|
B
i
|
∫
B
i
‍
|
f
(
y
)
|
γ
×
log
(
2
+
|
f
(
y
)
|
γ
)
d
y
}
)
q
+
C
i
q
(
1
+
2
i
r
ρ
(
x
0
)
)
-
3
N
0
|
B
i
|
-
1
ω
(
2
B
)
1
-
λ
(
∫
B
i
‍
|
f
(
y
)
|
t
d
y
)
q
≤
C
i
q
(
ω
(
2
B
)
ω
(
B
i
)
)
1
-
λ
ω
(
B
i
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
-
2
N
0
×
(
∫
B
i
‍
|
f
(
y
)
|
t
log
(
2
+
|
f
(
y
)
|
t
)
ω
(
y
)
1
/
q
d
y
)
q
+
C
i
q
(
ω
(
2
B
)
ω
(
B
i
)
)
1
-
λ
ω
(
B
i
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
-
2
N
0
×
(
∫
B
i
‍
|
f
(
y
)
|
t
ω
(
y
)
1
/
q
d
y
)
q
≤
C
i
q
2
-
i
n
(
1
-
λ
)
/
q
0
ω
(
B
i
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
α
×
(
∫
B
i
‍
|
f
(
y
)
|
t
log
(
2
+
|
f
(
y
)
|
t
)
a
a
a
a
a
a
×
ω
(
y
)
1
/
q
d
y
∫
B
i
‍
|
f
(
y
)
|
t
log
)
by taking N
0
=
[
-
α
+
(
l
0
+
n
+
4
)
q
θ
+
η
]
+
1 and N
=
4
N
0.
Hence,(76)
ω
(
2
B
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
α
×
ω
(
{
y
∈
B
(
x
0
,
r
)
:
|
[
b
,
ℐ
β
]
(
∑
i
=
1
∞
‍
f
i
)
(
y
)
|
>
t
}
)
≤
C
(
∑
i
=
1
∞
(
ω
(
2
B
)
-
λ
t
-
q
(
1
+
r
ρ
(
x
0
)
)
α
a
a
a
a
a
×
∫
B
(
x
0
,
r
)
‍
|
[
b
,
ℐ
β
]
f
i
(
y
)
ω
(
y
)
|
q
d
y
)
1
/
q
)
q
≤
C
(
∑
i
=
1
∞
‍
(
i
q
2
-
i
n
(
1
-
λ
)
/
q
0
ω
(
B
i
)
-
λ
(
1
+
r
ρ
(
x
0
)
)
α
a
a
a
a
a
×
(
∫
B
i
‍
|
f
(
y
)
|
t
log
(
2
+
|
f
(
y
)
|
t
)
a
a
a
a
a
a
a
a
×
ω
(
y
)
1
/
q
d
y
∫
B
i
‍
|
f
(
y
)
|
t
)
q
)
1
/
q
∑
i
=
1
∞
‍
)
q
≤
C
sup
B
(
x
0
,
r
)
⊂
ℝ
n
(
1
+
r
ρ
(
x
0
)
)
α
ω
(
2
B
)
-
λ
×
(
∫
B
(
x
0
,
r
)
‍
|
f
(
x
)
|
t
ln
(
2
+
|
f
(
y
)
|
t
)
a
a
a
a
a
a
a
×
ω
(
y
)
1
/
q
d
y
∫
B
(
x
0
,
r
)
‍
|
f
(
x
)
|
t
)
q
.
By (73) and (76), we obtain the desired result. This completes the proof.


## 5. The Calderón-Zygmund Inequality

For the open setΩ
⊂
ℝ
n,   ω
∈
A
p
ρ
,
∞
(
ℝ
n
)
(
1
≤
p
<
f
z
) and V
∈
B
n, we say f
∈
L
α
,
V
,
ω
p
,
λ
(
Ω
) if
(77)
∥
f
∥
L
α
,
V
p
,
λ
(
Ω
)
p
=
sup
B
(
x
0
,
r
)
⊂
ℝ
n
(
1
+
r
ρ
(
x
0
)
)
α
r
-
λ
×
∫
B
(
x
0
,
r
)
∩
Ω
‍
|
f
(
x
)
|
p
d
x
<
∞
.
In this section, we consider the behavior of the solution of the following Schrödinger equation:
(78)
(
-
Δ
+
V
)
u
=
f
(
x
)
,
a
.
e
.
x
∈
Ω
,
where f
∈
L
α
,
V
,
ω
p
,
λ
,
ω
(
Ω
), 1
<
p
<
∞,  0
<
λ
<
1,  α
∈
(
-
∞
,
∞
), and ω
∈
A
∞
ρ
,
∞.Theorem 16.
LetΩ be an open set in ℝ
n and α
∈
(
-
∞
,
∞
). If f
∈
L
α
,
V
,
ω
p
,
λ
(
Ω
), then there exists a function u
∈
L
α
,
V
,
ω
q
,
ν
λ
(
Ω
), where 1
<
p
<
n
/
2, 1
/
p
-
1
/
q
=
2
/
n, ν
=
q
/
p, 0
<
λ
<
1
/
ν, and ω
q
∈
A
1
+
q
/
p
′
ρ
,
∞, such that
(79)
(
-
Δ
+
V
)
u
=
f
(
x
)
,
a
.
e
.
x
∈
Ω
.
Furthermore,
(80)
∥
D
2
u
∥
L
α
,
V
,
ω
p
,
λ
(
Ω
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
Ω
)
,
where 1
<
p
<
∞, 0
<
λ
<
n, and ω
∈
A
p
ρ
,
∞;
(81)
∥
D
u
∥
L
α
,
V
,
ω
q
q
,
ν
1
λ
(
Ω
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
Ω
)
,
where 1
<
p
<
n
,
1
/
p
-
1
/
q
=
1
/
n
,
ν
1
=
q
/
p, 0
<
λ
<
n
/
ν
1, and ω
q
∈
A
1
+
q
/
p
′
ρ
,
∞;
(82)
∥
u
∥
L
α
,
V
,
ω
q
q
,
ν
λ
(
Ω
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
Ω
)
,
where 1
<
p
<
n
/
2, 1
/
p
-
1
/
q
=
2
/
n,  ν
=
q
/
p, 0
<
λ
<
1
/
ν, and ω
q
∈
A
1
+
q
/
p
′
ρ
,
∞.Proof.
From the proof of Theorem5, we have
(83)
∥
u
∥
L
α
,
V
,
ω
q
q
,
ν
λ
(
Ω
)
≤
C
∥
ℐ
2
f
∥
L
α
,
V
,
ω
q
q
,
ν
λ
(
Ω
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
Ω
)
.
From the proof of Theorem 1, we obtain
(84)
∥
D
2
u
∥
L
α
,
V
,
ω
p
,
λ
(
Ω
)
≤
C
∥
D
2
ℒ
-
1
f
∥
L
α
,
V
,
ω
p
,
λ
(
Ω
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
,
λ
(
Ω
)
.
Thus, (80) and (82) hold.
From page 543 in [1], we know
(85)
|
D
x
Γ
(
x
,
y
)
|
≤
C
k
(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
k
1
|
x
-
y
|
n
-
1
,
where Γ
(
x
,
y
) is the fundamental solution for ℒ
=
-
Δ
+
V.
Using (85) and adapting the argument for Theorem 5, we then have
(86)
∥
D
u
∥
L
α
,
V
,
ω
q
q
,
ν
1
λ
(
Ω
)
≤
C
∥
D
ℒ
-
1
f
∥
L
α
,
V
,
ω
q
q
,
ν
1
λ
(
Ω
)
≤
C
∥
f
∥
L
α
,
V
,
ω
p
p
,
λ
(
Ω
)
.
Thus, (81) holds. Hence, Theorem 16 is proved.



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*Source: 878629-2014-01-30.xml*