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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 961502, 20 pages
doi:10.1155/2010/961502
Research Article
Commutators of Littlewood-Paley Operators on the
Generalized Morrey Space
Yanping Chen,
1
Yong Ding,
2
and Xinxia Wang
3
1
Department of Mathematics and Mechanics, Applied Science School, University of Science and Technology
Beijing, Beijing 100083, China
2
Laboratory of Mathematics and Complex Systems (BNU), School of Mathematical Sciences, Beijing
Normal University, Ministry of Education, Beijing 100875, China
3
The College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang 830046, China
Correspondence should be addressed to Yanping Chen,
Received 6 May 2010; Accepted 11 July 2010
Academic Editor: Shusen Ding
Copyright q 2010 Yanping Chen et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Let μ
Ω
, μ
S
,andμ
∗,
λ
denote the Marcinkiewicz integral, the parameterized area integral, and
the parameterized Littlewood-Paley g
∗
λ
function, respectively. In this paper, the authors give a
characterization of BMO space by the boundedness of the commutators of μ
Ω
, μ
S
,andμ
∗,
λ
on
the generalized Morrey space L
p,ϕ
R
n
.
1. Introduction
Let S
n−1
{x ∈ R
n
: |x| 1} be the unit sphere in R
n
equipped with the Lebesgue measure
dσ. Suppose that Ω satisfies the following conditions.
aΩis the homogeneous function of degree zero on R
n
\{0},thatis,
Ω
μx
Ω
x
, for any μ>0,x∈ R
n
\
{
0
}
. 1.1
bΩhas mean zero on S
n−1
,thatis,
S
n−1
Ω
x
dσ
x
0.
1.2
2 Journal of Inequalities and Applications
cΩ∈ LipS
n−1
,thatis,
Ω
x
− Ω
y
≤
x
− y
, for any x
,y
∈ S
n−1
.
1.3
In 1958, Stein 1 defined the Marcinkiewicz integral of higher dimension μ
Ω
as
μ
Ω
f
x
∞
0
|
F
Ω,t
x
|
2
dt
t
3
1/2
,
1.4
where
F
Ω,t
x
|x−y|≤t
Ω
x − y
x − y
n−1
f
y
dy.
1.5
We refer to see 1, 2 for the properties of μ
Ω
.
Let 0 <<nand λ>1. The parameterized area integral μ
S
and the parameterized
Littlewood-Paley g
∗
λ
function μ
∗,
λ
are defined by
μ
S
f
x
⎛
⎝
Γ
x
1
t
|y−z|<t
Ω
y − z
y − z
n−
f
z
dz
2
dydt
t
n1
⎞
⎠
1/2
,
1.6
where Γx{y,t ∈ R
n1
: |x − y| <t}, and
μ
∗,
λ
f
x
⎛
⎝
R
n1
t
t
x − y
λn
1
t
|y−z|<t
Ω
y − z
y − z
n−
f
z
dz
2
dydt
t
n1
⎞
⎠
1/2
,
1.7
respectively. μ
S
and μ
∗,
λ
play very important roles in harmonic analysis and PDE e.g., see
3–8.
Before stating our result, let us recall some definitions. For b ∈ L
loc
R
n
, the
commutator b, μ
Ω
formed by b and the Marcinkiewicz integral μ
Ω
are defined by
b, μ
Ω
f
x
⎛
⎝
∞
0
|x−y|≤t
Ω
x − y
x − y
n−1
b
x
− b
y
f
y
dy
2
dt
t
3
⎞
⎠
1/2
.
1.8
Journal of Inequalities and Applications 3
Let 0 <<nand λ>1. The commutator b, μ
S
of μ
S
and the commutator b, μ
∗,
λ
of μ
∗,
λ
are defined, respectively, by
b, μ
S
f
x
⎛
⎝
Γ
x
1
t
|y−z|≤t
Ω
y − z
y − z
n−
b
x
− b
z
f
z
dz
2
dydt
t
n1
⎞
⎠
1/2
, 1.9
b, μ
∗,
λ
f
x
⎛
⎝
R
n1
t
t
x − y
λn
×
1
t
|y−z|≤t
Ω
y − z
y − z
n−
b
x
− b
z
f
z
dz
2
dydt
t
n1
⎞
⎠
1/2
.
1.10
Let b ∈ L
loc
R
n
.Itissaidthatb ∈ BMOR
n
if
b
∗
: sup
B⊂R
n
M
b, B
< ∞,
1.11
where B Bx, r denotes the ball in R
n
centered at x and with radius r,
M
b, B
1
|
B
|
B
|
b
x
− b
B
|
dx,
1.12
and b
B
1/|B|
B
bydy.
There are some results about the boundedness of the commutators formed by BMO
functions with μ
Ω
, μ
S
,andμ
∗,
λ
see 7, 9, 10.
Many important operators gave a characterization of BMO space. In 1976, Coifman et
al. 11 gave a characterization of BMO space by the commutator of Riesz transform; in 1982,
Chanillo 12 studied the commutator formed by Riesz potential and BMO and gave another
characterization of BMO space.
The purpose of this paper is to give a characterization of BMO space by the
boundedness of the commutators of μ
Ω
, μ
S
,andμ
∗,
λ
on the generalized Morrey space
L
p,ϕ
R
n
.
Definition 1.1. Let 1 <p<∞. Suppose that ϕ : 0, ∞ → 0, ∞ be such that ϕt is
nonincreasing and t
1/p
ϕt is nondecreasing. The generalized Morrey space L
p,ϕ
is defined
by
L
p,ϕ
R
n
f ∈ L
loc
R
n
:
f
L
p,ϕ
< ∞
, 1.13
where
f
L
p,ϕ
sup
x∈R
n
r>0
1
ϕ
|
B
x, r
|
1
|
B
x, r
|
Bx,r
f
y
p
dy
1/p
.
1.14
4 Journal of Inequalities and Applications
We refer to see 13, 14 for the known results of the generalized Morrey space L
p,ϕ
for some suitable ϕ.Notingthatϕt ≡ t
−1/p
, we get the Lebesque space L
p
R
n
. For ϕt
t
λ/n−1/p
0 <λ<n, L
p,ϕ
R
n
coincides with the Morrey space L
p,λ
R
n
.
The main result in this paper is as follows.
Theorem 1.2. Assume that ϕt is nonincreasing and t
1/p
ϕt is nondecreasing. Suppose that b, μ
Ω
is defined as 1.8, Ω satisfies 1.1, 1.2, and
Ω
x
− Ω
y
≤
C
1
log2/
x
− y
γ
,C
1
> 0,γ>1,x
,y
∈ S
n−1
.
1.15
If b, μ
Ω
is bounded on L
p,ϕ
R
n
for some p 1 <p<∞,thenb ∈ BMOR
n
.
Theorem 1.3. Let 0 <<nand 1 <p<∞. Assume that ϕt is nonincreasing and t
1/p
ϕt is
nondecreasing. Suppose that b, μ
S
is defined as 1.9, Ω satisfies 1.1, 1.2, and 1.15.Ifb, μ
S
is a bounded operator on L
p,ϕ
R
n
for some p 1 <p<∞,thenb ∈ BMOR
n
.
Theorem 1.4. Let 0 <<n, λ>1, and 1 <p<∞. Assume that ϕt is nonincreasing and
t
1/p
ϕt is nondecreasing. Suppose that b, μ
∗,ϕ
λ
is defined as 1.10, Ω satisfies 1.1, 1.2, and
1.15.Ifb, μ
∗,
λ
is on L
p,ϕ
R
n
for some p 1 <p<∞,thenb ∈ BMOR
n
.
Remark 1.5. It is easy to check that b, μ
S
fx ≤ 2
λn
b, μ
∗,
λ
fxsee, e.g., the proof
of 19 in 15, page 89, we therefore give only the proofs of Theorem 1.2 for b, μ
Ω
and
Theorem 1.3 for b, μ
S
.
Remark 1.6. It is easy to see that the condition 1.15 is weaker than Lip
β
S
n−1
for 0 <β≤ 1.
In the proof of Theorems 1.2 and 1.3, we will use some ideas in 16. However, because
Marcinkiewicz integral and the parameterized Littlewood-Paley operators are neither the
convolution operator nor the linear operators, hence, we need new ideas and nontrivial
estimates in the proof.
2. Proof of Theorem 1.2
Let us begin with recalling some known conclusion.
Similar to the proof of 17, we can easily get the following.
Lemma 2.1. If Ω satisfies conditions 1.1, 1.2, and 1.15,letβ>0, then for |x| > 2|y|, we have
Ω
x − y
x − y
β
−
Ω
x
|
x
|
β
≤
C
|
x
|
β
log
|
x
|
/
y
γ
.
2.1
Now let us return to the proof of Theorem 1.2. Suppose that b, μ
Ω
is a bounded
operator on L
p,ϕ
R
n
, we are going to prove that b ∈ BMOR
n
.
Journal of Inequalities and Applications 5
We may assume that b, μ
Ω
L
p,ϕ
→L
p,ϕ
1. We want to prove that, for any x
0
∈ R
n
and
r ∈ R
, the inequality
N
1
|
B
x
0
,r
|
Bx
0
,r
b
y
− a
0
dy ≤ A
p, Ω,n,γ
2.2
holds, where a
0
|Bx
0
,r|
−1
Bx
0
,r
bydy. Since b − a
0
,μ
Ω
b, μ
Ω
, we may assume that
a
0
0. Let
f
y
sgn
b
y
− c
0
χ
Bx
0
,r
y
, 2.3
where c
0
1/|Bx
0
,r|
Bx
0
,r
sgnbydy. Since 1/|Bx
0
,r|
Bx
0
,r
bydy a
0
0, we
can easily get |c
0
| < 1. Then, f has the following properties:
f
∞
≤ 2, 2.4
supp f ⊂ B
x
0
,r
, 2.5
R
n
f
y
dy 0, 2.6
f
y
b
y
> 0,y∈ B
x
0
,r
, 2.7
1
|
B
x
0
,r
|
R
n
f
y
b
y
dy N.
2.8
In this proof for j 1, ,15, A
j
is a positive constant depending only on Ω,p, n, γ,and
A
i
1 ≤ i<j. Since Ω satisfies 1.2, then there exists an A
1
such that 0 <A
1
< 1and
σ
x
∈ S
n−1
: Ω
x
≥
2C
1
log2/A
1
γ
> 0, 2.9
where σ is the measure on S
n−1
which is induced from the Lebesgue measure on R
n
.Bythe
condition 1.15,itiseasytoseethat
Λ :
x
∈ S
n−1
: Ω
x
≥
2C
1
log2/A
1
γ
2.10
is a closed set. We claim that
if x
∈ Λ and y
∈ S
n−1
, satisfying
x
− y
≤ A
1
, then Ω
y
≥
C
1
log2/A
1
γ
.
2.11
6 Journal of Inequalities and Applications
In fact, since |Ωx
− Ωy
|≤C
1
/log2/|x
− y
|
γ
≤ C
1
/log2/A
1
γ
, note that Ωx
≥
2C
1
/log2/A
1
γ
, we can get Ωy
≥ C
1
/log2/A
1
γ
. Taking A
2
> 3/A
1
,let
G
x ∈ R
n
:
|
x − x
0
|
≥ A
2
r,
x − x
0
∈ Λ
. 2.12
For x ∈ G, we have
b, μ
Ω
f
x
≥
μ
Ω
bf
x
−
|
b
x
|
μ
Ω
f
x
⎧
⎪
⎨
⎪
⎩
∞
0
|x−y|≤t
Ω
x − y
x − y
n−1
b
y
f
y
dy
2
dt
t
3
⎫
⎪
⎬
⎪
⎭
1/2
−
|
b
x
|
⎧
⎪
⎨
⎪
⎩
∞
0
|x−y|≤t
Ω
x − y
x − y
n−1
f
y
dy
2
dt
t
3
⎫
⎪
⎬
⎪
⎭
1/2
: I
1
− I
2
.
2.13
For I
1
, noting that if y ∈ Bx
0
,r, then |x − x
0
| >A
2
|y − x
0
| for x ∈ G. Thus, we have
x − y
−
x − x
0
≤ 2
y − x
0
|
x − x
0
|
≤
2
A
2
<A
1
.
2.14
Using 2.11,wegetΩx − y
≥ C
1
/log2/A
1
γ
. Noting that |x − x
0
||x − y|, it follows
from 2.5, 2.7, 2.8,andH
¨
older’s inequality that
I
1
≥
⎧
⎪
⎨
⎪
⎩
∞
|
x−x
0
|
⎛
⎜
⎝
Bx
0
,r
Ω
x − y
b
y
f
y
x − y
n−1
χ
{|x−y|≤t}
y
dy
⎞
⎟
⎠
2
dt
t
3
⎫
⎪
⎬
⎪
⎭
1/2
≥
⎛
⎜
⎝
∞
|
x−x
0
|
Bx
0
,r
Ω
x − y
b
y
f
y
x − y
n−1
χ
{|x−y|≤t}
dy
dt
t
3
⎞
⎟
⎠
∞
|
x−x
0
|
dt
t
3
−1/2
≥
C
1
log2/A
1
γ
|
x − x
0
|
Bx
0
,r
x − y
−n1
b
y
f
y
|x−x
0
|≤t
|x−y|≤t
dt
t
3
dy
≥
C
log2/A
1
γ
|
x − x
0
|
−n
Bx
0
,r
b
y
f
y
dy
A
3
Nr
n
|
x − x
0
|
−n
.
2.15
Journal of Inequalities and Applications 7
For x ∈ G,byΩ ∈ L
∞
S
n−1
, 2.4, 2.5, 2.6, the Minkowski inequality, and Lemma 2.1,we
obtain
I
2
|
b
x
|
⎧
⎨
⎩
∞
0
R
n
f
y
Ω
x − y
x − y
n−1
χ
{|x−y|≤t}
−
Ω
x − x
0
|
x − x
0
|
n−1
χ
{|x−x
0
|≤t}
dy
2
dt
t
3
⎫
⎬
⎭
1/2
≤
|
b
x
|
⎧
⎪
⎨
⎪
⎩
⎛
⎝
∞
0
|x−y|≤t<|x−x
0
|
Ω
x − y
x − y
n−1
f
y
dy
2
dt
t
3
⎞
⎠
1/2
⎛
⎝
∞
0
|x−x
0
|≤t<|x−y|
|
Ω
x − x
0
|
|
x − x
0
|
n−1
f
y
dy
2
dt
t
3
⎞
⎠
1/2
⎛
⎜
⎝
∞
0
⎛
⎝
|x−x
0
|≤t
|x−y|≤t
Ω
x − y
x − y
n−1
−
Ω
x − x
0
|
x − x
0
|
n−1
f
y
dy
⎞
⎠
2
dt
t
3
⎞
⎟
⎠
1/2
⎫
⎪
⎪
⎬
⎪
⎪
⎭
≤
|
b
x
|
⎧
⎨
⎩
Bx
0
,r
Ω
x − y
x − y
n−1
f
y
|x−y|≤t<|x−x
0
|
dt
t
3
1/2
dy
Bx
0
,r
|
Ω
x − x
0
|
|
x − x
0
|
n−1
f
y
|x−y|>t≥|x−x
0
|
dt
t
3
1/2
dy
Bx
0
,r
f
y
Ω
x − y
x − y
n−1
−
Ω
x − x
0
|
x − x
0
|
n−1
⎛
⎝
|x−y|≤t
|x−x
0
|≤t
dt
t
3
⎞
⎠
1/2
dy
⎫
⎪
⎬
⎪
⎭
≤ C
|
b
x
|
r
1/2
Bx
0
,r
f
y
|
x − x
0
|
n1/2
dy
Bx
0
,r
f
y
|
x − x
0
|
n
log
|
x − x
0
|
/r
γ
dy
≤ A
4
|
b
x
|
r
n
|
x − x
0
|
−n
log
|
x − x
0
|
r
−γ
.
2.16
Let
F
x ∈ G :
|
b
x
|
>
A
3
N
2A
4
log
|
x − x
0
|
r
γ
,
|
x − x
0
|
<N
1/n
r
.
2.17
Without loss of generality, we may assume that N>A
2
> 1, otherwise, we get the desired
8 Journal of Inequalities and Applications
result. Since ϕt is nonincreasing, it follows that ϕ|Bx
0
,N
1/n
r| ≤ ϕ|Bx
0
,r|ϕr
n
.By
2.13, 2.15,and2.16, we have
f
p
L
p,ϕ
≥
b, μ
Ω
f
p
L
p,ϕ
≥
1
ϕ
B
x
0
,N
1/n
r
p
B
x
0
,N
1/n
r
|x−x
0
|<N
1/n
r
b, μ
Ω
f
x
p
dx
≥
1
ϕ
r
n
p
Nr
n
G\F∩{|x−x
0
|<N
1/n
r}
1
2
A
3
Nr
n
|
x − x
0
|
−n
p
dx
≥
1
ϕ
r
n
p
Nr
n
{A
5
|F|A
2
r
n
1/n
<|x−x
0
|<N
1/n
r}∩G
1
2
A
3
Nr
n
|
x − x
0
|
−n
p
dx
ω
n−1
ϕ
r
n
p
Nr
n
A
3
Nr
n
2
p
N
1/n
r
A
5
|
F
|
A
2
r
n
1/n
t
−pnn−1
dt
≥
ω
n−1
ϕ
r
n
p
Nr
n
p−1
A
3
/2
p
n − np
N
1−p
r
n1−p
− A
1−pn
5
|
F
|
A
2
r
n
1−p
.
2.18
Thus,
|
F
|
A
2
r
n
1−p
≤ A
6
N
1−p
r
n1−p
1 ϕ
r
n
p
f
p
L
p,ϕ
. 2.19
Now, we claim that
f
L
p,ϕ
≤
C
ϕ
r
n
,
2.20
where C is independent of r. In fact,
f
L
p,ϕ
sup
x∈R
n
t>0
1
ϕ
|
B
x, t
|
1
|
B
x, t
|
Bx,t
f
y
p
dy
1/p
.
2.21
Now, we consider the L
p,ϕ
norm of f in the following two cases.
Case 1 t>r. Since s
1/p
ϕs is nondecreasing in s, then
1
ϕ
|
B
x, t
|
1
|
B
x, t
|
1/p
≤
1
ϕ
r
n
1
r
n/p
.
2.22
Journal of Inequalities and Applications 9
Thus,
f
L
p,ϕ
≤ sup
x∈R
n
t>0
1
ϕ
r
n
1
r
n/p
Bx,t
f
y
p
dy
1/p
sup
x∈R
n
t>0
1
ϕ
r
n
1
r
n/p
Bx,t∩Bx
0
,r
f
y
p
dy
1/p
≤
C
ϕ
r
n
.
2.23
Case 2 t ≤ r. Since ϕs is nonincreasing in s, then
1
ϕ
|
B
x, t
|
≤
1
ϕ
r
n
.
2.24
Thus,
f
L
p,ϕ
≤ sup
x∈R
n
t>0
1
ϕ
r
n
1
|
B
x, t
|
Bx,t
f
y
p
dy
1/p
≤
C
ϕ
r
n
.
2.25
Now, 2.20 is established. Then, by 2.19 and 2.20,weget
|
F
|
A
2
r
n
≥ A
7
Nr
n
. 2.26
If N ≤ 2A
−1
7
A
n
2
, then Theorem 1.2 is proved. If N>2A
−1
7
A
n
2
, then
|
F
|
≥
A
7
2
Nr
n
.
2.27
Let gyχ
Bx
0
,r
y. For x ∈ F, we have
b, μ
Ω
g
x
≥
|
b
x
|
⎧
⎪
⎨
⎪
⎩
∞
0
|x−y|≤t
Ω
x − y
x − y
n−1
g
y
dy
2
dt
t
3
⎫
⎪
⎬
⎪
⎭
1/2
−
⎧
⎪
⎨
⎪
⎩
∞
0
|x−y|≤t
Ω
x − y
x − y
n−1
b
y
g
y
dy
2
dt
t
3
⎫
⎪
⎬
⎪
⎭
1/2
: K
1
− K
2
.
2.28
10 Journal of Inequalities and Applications
Noting that if y ∈ Bx
0
,r and x ∈ F,weget|x − y
− x − x
0
|≤A
1
. Applying 2.11,we
have Ωx − y
≥ C
1
/log2/A
1
γ
. Since |x − y||x − x
0
| when y ∈ Bx
0
,r and x ∈ F,it
follows that
K
1
≥
|
b
x
|
⎧
⎪
⎨
⎪
⎩
∞
|
x−x
0
|
⎛
⎜
⎝
Bx
0
,r
Ω
x − y
x − y
n−1
χ
{|x−y|≤t}
y
dy
⎞
⎟
⎠
2
dt
t
3
⎫
⎪
⎬
⎪
⎭
1/2
≥
|
b
x
|
⎛
⎜
⎝
∞
|
x−x
0
|
Bx
0
,r
Ω
x − y
x − y
n−1
χ
{|x−y|≤t}
dy
dt
t
3
⎞
⎟
⎠
∞
|
x−x
0
|
dt
t
3
−1/2
≥
C
1
|
b
x
|
log2/A
1
γ
|
x − x
0
|
Bx
0
,r
x − y
−n1
|x−x
0
|≤t
|x−y|≤t
dt
t
3
dy
≥ A
8
|
b
x
||
x − x
0
|
−n
Bx
0
,r
dy
A
8
r
n
|
b
x
||
x − x
0
|
−n
.
2.29
By Ω ∈ L
∞
S
n−1
, |x− x
0
||x −y| when y ∈ Bx
0
,r and x ∈ F and the Minkowski inequality,
we have
K
2
≤ C
Bx
0
,r
b
y
x − y
n
dy
≤ A
9
|
x − x
0
|
−n
Bx
0
,r
b
y
dy
A
9
Nr
n
|
x − x
0
|
−n
.
2.30
Thus, by 2.28, 2.29,and2.30,weget,forx ∈ F,
b, μ
Ω
g
x
≥ A
8
r
n
|
b
x
||
x − x
0
|
−n
− A
9
Nr
n
|
x − x
0
|
−n
. 2.31
Similar to the proof of 2.20, we can easily get g
L
p,ϕ
≤ C/ϕr
n
.Thus,by2.31,
Journal of Inequalities and Applications 11
ϕNr
n
≤ ϕr
n
,and|bx| > NA
3
/2A
4
log|x − x
0
|/r
γ
when x ∈ F, we have
A
10
ϕ
r
n
≥
g
L
p,ϕ
≥
b, μ
Ω
g
L
p,ϕ
≥
1
ϕ
Nr
n
Nr
n
1/p
|x−x
0
|<N
1/n
r
b, μ
Ω
g
x
p
dx
1/p
≥
1
ϕ
r
n
Nr
n
1/p
|x−x
0
|<N
1/n
r
b, μ
Ω
g
x
dx
|x−x
0
|<N
1/n
r
dx
−1/p
≥
1
ϕ
r
n
Nr
n
F
b, μ
Ω
g
x
dx
≥
A
8
r
n
ϕ
r
n
Nr
n
F
|
b
x
||
x − x
0
|
−n
dx −
A
9
Nr
n
ϕ
r
n
Nr
n
F
|
x − x
0
|
−n
dx
≥
A
11
ϕ
r
n
F
log
|
x − x
0
|
r
γ
|
x − x
0
|
−n
dx
−
A
9
ϕ
r
n
F
|
x − x
0
|
−n
dx
: L
1
− L
2
.
2.32
We first estimate L
2
. Since A
2
r<|x − x
0
| <N
1/n
r for x ∈ F, we have
L
2
≤
A
9
ω
n−1
ϕ
r
n
N
1/n
r
A
2
r
ρ
−1
dρ ≤
A
12
ϕ
r
n
log N.
2.33
Now, the estimate of L
1
is divided into two cases, namely, 1: γ ≥ n;2:1<γ<n.
Case 1 γ ≥ n. Since the function log s/s is decreasing for s ≥ 3and3r<A
2
r<|x − x
0
| <
N
1/n
r for x ∈ F, by 2.27,weget
L
1
A
11
r
−n
ϕ
r
n
F
log
|
x − x
0
|
/r
|
x − x
0
|
/r
n
log
|
x − x
0
|
r
γ−n
dx
≥
A
7
A
11
2ϕ
r
n
log A
2
γ−n
N
log N
1/n
N
1/n
n
≥
A
13
ϕ
r
n
log N
n
.
2.34
12 Journal of Inequalities and Applications
Case 2 1 <γ<n. Since the function log s
γ
/s
n
is decreasing for s ≥ 3and3r<A
2
r<
|x − x
0
| <N
1/n
r for x ∈ F,by2.27, we have
L
1
A
11
r
−n
ϕ
r
n
F
log
|
x − x
0
|
/r
γ
|
x − x
0
|
/r
n
dx
≥
A
7
A
11
2ϕ
r
n
N
log N
1/n
γ
N
≥
A
14
ϕ
r
n
log N
γ
.
2.35
From Cases 1 and 2, we know that there exists a constant τ>1 such that
L
1
≥
A
15
ϕ
r
n
log N
τ
.
2.36
So by 2.32, 2.33,and2.36,weget
A
10
≥ A
15
log N
τ
− A
12
log N.
2.37
Then, N ≤ AΩ,p,n,γ. Theorem 1.2 is proved.
3. Proof of Theorem 1.3
Similar to the proof of Theorem 1.2, we only give the outline.
Suppose that b, μ
S
is a bounded operator on L
p,ϕ
R
n
, we are going to prove that
b ∈ BMOR
n
.
We may assume that b, μ
S
L
p,ϕ
→L
p,ϕ
1. We want to prove that, for any x
0
∈ R
n
and
r ∈ R
, the inequality
N
1
|
B
x
0
,r
|
Bx
0
,r
b
y
− a
0
dy ≤ B
Ω,p,n,
3.1
holds, where a
0
|Bx
0
,r|
−1
Bx
0
,r
bydy. Since b − a
0
,μ
S
b, μ
S
, we may assume that
a
0
0. Let fy be as 2.3, then 2.4–2.8 hold. In this proof for j 1, ,13, B
j
is a positive
constant depending only on Ω,p, n, ,andB
i
1 ≤ i<j. Since Ω satisfies 1.2, then there
exists a B
1
such that 0 <B
1
< 1and
σ
x
∈ S
n−1
: Ω
x
≥
2C
1
log2/B
1
γ
> 0, 3.2
Journal of Inequalities and Applications 13
where σ is the measure on S
n−1
which is induced from the Lebesgue measure on R
n
.Bythe
condition 1.15,itiseasytoseethat
Λ :
x
∈ S
n−1
: Ω
x
≥
2C
1
log2/B
1
γ
3.3
is a closed set. As the proof of 2.11, we can get the following:
if x
∈ Λ and y
∈ S
n−1
, satisfying
x
− y
≤ B
1
, then Ω
y
≥
C
1
log2/B
1
γ
.
3.4
Taking B
2
> 3/B
1
1, let
G
x ∈ R
n
:
|
x − x
0
|
≥ B
2
r,
x − x
0
∈ Λ
. 3.5
For x ∈ G, we have
b, μ
S
f
x
⎛
⎝
∞
0
|x−y|<t
|y−z|<t
Ω
y − z
y − z
n−
b
x
− b
z
f
z
dz
2
dydt
t
n12
⎞
⎠
1/2
≥
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
×
|y−z|<t
Ω
y − z
y − z
n−
b
x
− b
z
f
z
dz
2
dydt
t
n12
⎞
⎠
1/2
≥
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|, y−x
0
∈Λ
×
|y−z|<t
Ω
y − z
y − z
n−
b
z
f
z
dz
2
dydt
t
n12
⎞
⎠
1/2
−
|
b
x
|
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
|y−z|<t
Ω
y − z
y − z
n−
f
z
dz
2
dydt
t
n12
⎞
⎠
1/2
: I
1
− I
2
.
3.6
14 Journal of Inequalities and Applications
For I
1
, noting that if |z − x
0
| <r, |x − x
0
| >B
2
|z − x
0
|,and|y − x
0
| > 2B
2
|z − x
0
|, then we get
y − z
−
y − x
0
≤ 2
|
z − x
0
|
y − x
0
≤
1
B
2
<B
1
.
3.7
Then by 3.4,wegetΩy − z
≥ C
1
/log2/B
1
γ
. Since 4|x−x
0
| > |y−x
0
||z−x
0
|≥|y−z|≥
|y−x
0
|−|z−x
0
| > 2|x−x
0
|−|x−x
0
|/2 3|x−x
0
|/2and4|x−x
0
| > |x−y|≥|y−x
0
|−|x−x
0
| > |x−x
0
|,
we get 4|x − x
0
|≥|y − z|≥3|x − x
0
|/2and4|x − x
0
| > |x − y| > |x − x
0
|.Thus,by2.5, 2.7,
2.8,andtheH
¨
older inequality, we get
I
1
≥ C
∞
4
|
x−x
0
|
|x−y|<t, y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
Bx
0
,r
Ω
y − z
y − z
n−
b
z
f
z
χ
{|y−z|<t}
dz
dydt
t
n12
×
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t, y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
dydt
t
n12
⎞
⎠
−1/2
≥ C
|
x − x
0
|
2−n
Bx
0
,r
b
z
f
z
y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
4|x−x
0
|<t, |x−y|<t
|y−z|<t
dtdy
t
n12
dz
C
|
x − x
0
|
2−n
Bx
0
,r
b
z
f
z
y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
4|x−x
0
|<t
dtdy
t
n12
dz
≥ C
|
x − x
0
|
−n
Bx
0
,r
b
z
f
z
dz
B
3
Nr
n
|
x − x
0
|
−n
.
3.8
By 2.5 and 2.6, we have
I
2
|
b
x
|
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
×
R
n
Ω
y − z
y − z
n−
χ
{|y−z|<t}
−
Ω
y − x
0
y − x
0
n−
χ
{|y−x
0
|<t}
f
z
dz
2
dydt
t
n12
⎞
⎠
1/2
Journal of Inequalities and Applications 15
≤
|
b
x
|
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
×
|y−z|<t
|y−x
0
|<t
Ω
y − z
y − z
n−
−
Ω
y − x
0
y − x
0
n−
f
z
dz
2
dydt
t
n12
⎞
⎟
⎠
1/2
|
b
x
|
⎛
⎜
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
|y−z|<t
|y−x
0
|≥t
Ω
y − z
y − z
n−
f
z
dz
2
dydt
t
n12
⎞
⎟
⎠
1/2
|
b
x
|
⎛
⎜
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
|y−z|≥t
|y−x
0
|<t
Ω
y − x
0
y − x
0
n−
f
z
dz
2
dydt
t
n12
⎞
⎟
⎠
1/2
: I
1
2
I
2
2
I
3
2
.
3.9
In I
2
2
, we have t ≤|y − x
0
| < 3|x − x
0
| and t ≥ 4|x − x
0
|.InI
3
2
, we get t ≤|y − z| < 4|x − x
0
|
and t ≥ 4|x − x
0
|. It is easy to see that I
2
2
I
3
2
0. Now, we estimate I
1
2
, by Ω ∈ L
∞
S
n−1
,the
Minkowski inequality, Lemma 2.1 for |y − x
0
| > 2|z − x
0
|,and2.4,weget
I
1
2
≤ C
|
b
x
|
Bx
0
,r
f
z
dz
⎛
⎝
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
4|x−x
0
|≤t, |y−z|<t
|y−x
0
|<t, |x−y|≤t
×
1
y − x
0
2n−
log
y − x
0
/r
2γ
dtdy
t
n12
1/2
≤ B
4
|
b
x
|
r
n
|
x − x
0
|
−n
log
|
x − x
0
|
r
−γ
.
3.10
From 3.9 and 3.10,weget
I
2
≤ B
4
|
b
x
|
r
n
|
x − x
0
|
−n
log
|
x − x
0
|
r
−γ
.
3.11
Let
F
x ∈ G :
|
b
x
|
>
B
3
N
2B
4
log
|
x − x
0
|
r
γ
,
|
x − x
0
|
<N
1/n
r
.
3.12
16 Journal of Inequalities and Applications
Without loss of generality, we may assume that N>B
2
> 1, otherwise, we get the desired
result. Since ϕt is nonincreasing, we have ϕ|Bx
0
,N
1/n
r| ≤ ϕ|Bx
0
,r|ϕr
n
. Then by,
3.6, 3.8,and3.11,weget
f
p
L
p,ϕ
≥
b, μ
S
f
p
L
p,ϕ
≥
1
ϕ
B
x
0
,N
1/n
r
p
B
x
0
,N
1/n
r
|x−x
0
|<N
1/n
r
b, μ
S
f
x
p
dx
≥
1
ϕ
r
n
p
Nr
n
G\F∩{|x−x
0
|<N
1/n
r}
1
2
B
3
Nr
n
|
x − x
0
|
−n
p
dx
≥
1
ϕ
r
n
p
Nr
n
{B
5
|F|B
2
r
n
1/n
<|x−x
0
|<N
1/n
r}∩G
1
2
B
3
Nr
n
|
x − x
0
|
−n
p
dx
1
ϕ
r
n
p
Nr
n
B
3
Nr
n
2
p
N
1/n
r
B
5
|
F
|
B
2
r
n
1/n
t
−pnn−1
dt
Λ
J
x
dσ
x
≥
1
ϕ
r
n
p
σ
Λ
Nr
n
p−1
B
3
/2
p
n − np
N
1−p
r
n1−p
− B
1−pn
5
|
F
|
B
2
r
n
1−p
.
3.13
Thus,
|
F
|
B
2
r
n
1−p
≤ B
6
N
1−p
r
n1−p
1
ϕ
r
n
p
f
p
L
p,λ
. 3.14
Then, by 2.20 and 3.14,weget
|
F
|
B
2
r
n
≥ B
7
Nr
n
. 3.15
If N ≤ 2B
−1
7
B
n
2
, then Theorem 1.3 is proved. If N>2B
−1
7
B
n
2
, then
|
F
|
≥
B
7
2
Nr
n
.
3.16
Journal of Inequalities and Applications 17
Let gyχ
Bx
0
,r
y. For x ∈ F, we have
b, μ
S
g
x
⎛
⎝
∞
0
|x−y|<t
|y−z|<t
Ω
y − z
y − z
n−
b
x
− b
z
g
z
dz
2
dydt
t
n12
⎞
⎠
1/2
≥
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
|y−z|<t
Ω
y−z
y−z
n−
b
x
−b
z
g
z
dz
2
dydt
t
n12
⎞
⎠
1/2
≥
|
b
x
|
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|, y−x
0
∈Λ
|y−z|<t
Ω
y−z
y−z
n−
g
z
dz
2
dydt
t
n12
⎞
⎠
1/2
−
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
|y−z|<t
Ω
y − z
y − z
n−
b
z
g
z
dz
2
dydt
t
n12
⎞
⎠
1/2
: K
1
− K
2
.
3.17
For K
1
, as above mentioned, we have Ωy − z
≥ C
1
/log2/B
1
γ
. Since 4|x− x
0
|≥|y −z|≥
3|x − x
0
|/2and4|x − x
0
| > |x − y| > |x − x
0
|, it follows the H
¨
older inequality that
K
1
|
b
x
|
⎧
⎨
⎩
∞
4
|
x−x
0
|
|x−y|<t, y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
Bx
0
,r
Ω
y−z
y−z
n−
χ
{|y−z|<t }
dz
2
dydt
t
n12
⎫
⎬
⎭
1/2
≥
|
b
x
|
∞
4
|
x−x
0
|
|x−y|<t, y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
Bx
0
,r
Ω
y − z
y − z
n−
χ
{|y−z|<t}
dz
dydt
t
n12
×
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t, y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
dydt
t
n12
⎞
⎠
−1/2
≥ C
|
b
x
||
x − x
0
|
2−n
Bx
0
,r
y−x
0
∈Λ
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
4|x−x
0
|<t
dtdy
t
n12
dz
≥ B
8
N
|
x − x
0
|
−n
Bx
0
,r
dz
B
8
Nr
n
|
x − x
0
|
−n
.
3.18
18 Journal of Inequalities and Applications
By Ω ∈ L
∞
S
n−1
, the Minkowski inequality, and |x−x
0
||y−z| for 2|x−x
0
| < |y−x
0
| < 3|x−x
0
|
and z ∈ Bx
0
,r,weget
K
2
⎛
⎝
∞
4
|
x−x
0
|
|x−y|<t
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
Bx
0
,r
Ω
y − z
y − z
n−
b
z
χ
{|y−z|<t}
dz
2
dydt
t
n12
⎞
⎠
1/2
≤
C
|
x − x
0
|
n−
Bx
0
,r
|
b
z
|
dz
2|x−x
0
|<|y−x
0
|<3|x−x
0
|
∞
4
|
x−x
0
|
dtdy
t
n12
1/2
≤ B
9
|
x − x
0
|
−n
Bx
0
,r
|
b
z
|
dz
B
9
Nr
n
|
x − x
0
|
−n
.
3.19
Thus, by 3.17, 3.18,and3.19,weget,forx ∈ F,
b, μ
S
g
x
≥ B
8
r
n
|
b
x
||
x − x
0
|
−n
− B
9
Nr
n
|
x − x
0
|
−n
. 3.20
Thus, by 3.20, ϕNr
n
≤ ϕr
n
, |bx| > NB
3
/2B
4
log|x − x
0
|/r
γ
when x ∈ F and the
H
¨
older inequality, we have
B
10
ϕ
r
n
≥
g
L
p,ϕ
≥
b, μ
S
g
L
p,ϕ
≥
1
ϕ
Nr
n
Nr
n
1/p
|x−x
0
|<N
1/n
r
b, μ
S
g
x
p
dx
1/p
≥
1
ϕ
r
n
Nr
n
1/p
|x−x
0
|<N
1/n
r
b, μ
S
g
x
dx
|x−x
0
|<N
1/n
r
dx
−1/p
≥
1
ϕ
r
n
Nr
n
F
b, μ
S
g
x
dx
≥
B
8
r
n
ϕ
r
n
Nr
n
F
|
b
x
||
x − x
0
|
−n
dx −
B
9
Nr
n
ϕ
r
n
Nr
n
F
|
x − x
0
|
−n
dx
≥
B
11
ϕ
r
n
F
log
|
x − x
0
|
r
γ
|
x − x
0
|
−n
dx
−
B
9
ϕ
r
n
F
|
x − x
0
|
−n
dx
: L
1
− L
2
.
3.21
Journal of Inequalities and Applications 19
As the proof of 2.33 and 2.36, we can get that there exists a constant τ>1 such that
L
1
≥
B
12
ϕ
r
n
log N
τ
,
L
2
≤
B
13
ϕ
r
n
log N.
3.22
So, by 3.21 and 3.22,weget
B
10
≥ B
12
log N
τ
− B
13
log N.
3.23
Then, N ≤ BΩ,p,n,. Theorem 1.3 is proved.
Acknowledgments
The authors wish to express their gratitude to the referee for his/her valuable comments and
suggestions. The research was supported by NSF of China Grant nos.: 10901017, 10931001,
SRFDPofChinaGrant no.: 20090003110018, and NSF of Zhenjiang Grant no.: Y7080325.
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