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 Wang3
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, yanpingch@126.com
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 μΩ , μ
denote the Marcinkiewicz integral, the parameterized area integral, and
S , and μλ
the parameterized Littlewood-Paley gλ∗ function, respectively. In this paper, the authors give a
∗,
characterization of BMO space by the boundedness of the commutators of μΩ , μ
on
S , and μλ
p,ϕ
n
the generalized Morrey space L R .
1. Introduction
Let Sn−1 {x ∈ Rn : |x| 1} be the unit sphere in Rn equipped with the Lebesgue measure
dσ. Suppose that Ω satisfies the following conditions.
a Ω is the homogeneous function of degree zero on Rn \ {0}, that is,
Ω μx Ωx,
for any μ > 0, x ∈ Rn \ {0}.
1.1
b Ω has mean zero on Sn−1 , that is,
Ω x dσ x 0.
Sn−1
1.2
2
Journal of Inequalities and Applications
c Ω ∈ LipSn−1 , that is,
Ω x − Ω y ≤ x − y ,
for any x , y ∈ Sn−1 .
1.3
In 1958, Stein 1 defined the Marcinkiewicz integral of higher dimension μΩ as
μΩ f x ∞
0
dt
|FΩ,t x|2 3
t
1/2
,
1.4
where
FΩ,t x |x−y|≤t
Ω x−y
f y dy.
x − yn−1
1.5
We refer to see 1, 2 for the properties of μΩ .
Let 0 < < n and λ > 1. The parameterized area integral μ
S and the parameterized
are
defined
by
Littlewood-Paley gλ∗ function μ∗,
λ
⎛
⎝
μ
S fx ⎞1/2
2
dydt
1
Ω
y
−
z
n− fzdz n1 ⎠ ,
t
t
y
−
z
Γx
|y−z|<t
1.6
: |x − y| < t}, and
where Γx {y, t ∈ Rn1
⎛
μ∗,
fx ⎝
λ
Rn1
t
t x − y
⎞1/2
2
λn dydt
1
Ω
y
−
z
n− fzdz n1 ⎠ ,
t
t
|y−z|<t y − z
1.7
∗,
play very important roles in harmonic analysis and PDE e.g., see
respectively. μ
S and μλ
3–8.
Before stating our result, let us recall some definitions. For b ∈ Lloc Rn , the
commutator b, μΩ formed by b and the Marcinkiewicz integral μΩ are defined by
b, μΩ
⎛
∞ fx ⎝
0 |x−y|≤t
2 ⎞1/2
Ω x−y dt
bx − b y f y dy 3 ⎠ .
t
x − yn−1
1.8
Journal of Inequalities and Applications
3
∗,
∗,
Let 0 < < n and λ > 1. The commutator b, μ
S of μS and the commutator b, μλ of μλ
are defined, respectively, by
b, μ
S
⎛
fx ⎝
⎛
⎝
fx
b, μ∗,
λ
1
Γx t
|y−z|≤t
Rn1
⎞1/2
2
dydt
Ω y−z
bx − bzfzdz n1 ⎠ ,
t
y − zn−
λn
t
t x − y
1
× t
|y−z|≤t
1.9
⎞1/2
2
dydt
Ω y−z
bx − bzfzdz n1 ⎠ .
t
y − zn−
1.10
Let b ∈ Lloc Rn . It is said that b ∈ BMORn if
b∗ : sup Mb, B < ∞,
1.11
B⊂Rn
where B Bx, r denotes the ball in Rn centered at x and with radius r,
Mb, B 1
|B|
|bx − bB |dx,
1.12
B
and bB 1/|B| B bydy.
There are some results about the boundedness of the commutators formed by BMO
∗,
see 7, 9, 10.
functions with μΩ , μ
S , and μλ
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
∗,
on the generalized Morrey space
boundedness of the commutators of μΩ , μ
S , and μλ
p,ϕ
n
L R .
Definition 1.1. Let 1 < p < ∞. Suppose that ϕ : 0, ∞ → 0, ∞ be such that ϕt is
nonincreasing and t1/p ϕt is nondecreasing. The generalized Morrey space Lp,ϕ is defined
by
Lp,ϕ Rn f ∈ Lloc Rn : f Lp,ϕ < ∞ ,
1.13
where
f p,ϕ sup
L
1
n ϕ|Bx, r|
x∈R
r>0
1
|Bx, r|
p
f y dy
Bx,r
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 Lp,ϕ
for some suitable ϕ. Noting that ϕt ≡ t−1/p , we get the Lebesque space Lp Rn . For ϕt tλ/n−1/p 0 < λ < n, Lp,ϕ Rn coincides with the Morrey space Lp,λ Rn .
The main result in this paper is as follows.
Theorem 1.2. Assume that ϕt is nonincreasing and t1/p ϕt is nondecreasing. Suppose that b, μΩ is defined as 1.8, Ω satisfies 1.1, 1.2, and
Ω x − Ω y ≤ C1
γ ,
log2/x − y C1 > 0, γ > 1, x , y ∈ Sn−1 .
1.15
If b, μΩ is bounded on Lp,ϕ Rn for some p 1 < p < ∞, then b ∈ BMORn .
Theorem 1.3. Let 0 < < n and 1 < p < ∞. Assume that ϕt is nonincreasing and t1/p ϕt is
nondecreasing. Suppose that b, μ
S is defined as 1.9, Ω satisfies 1.1, 1.2, and 1.15. If b, μS is a bounded operator on Lp,ϕ Rn for some p 1 < p < ∞, then b ∈ BMORn .
Theorem 1.4. Let 0 < < n, λ > 1, and 1 < p < ∞. Assume that ϕt is nonincreasing and
∗,ϕ
t1/p ϕt is nondecreasing. Suppose that b, μλ is defined as 1.10, Ω satisfies 1.1, 1.2, and
∗,
p,ϕ
n
1.15. If b, μλ is on L R for some p 1 < p < ∞, then b ∈ BMORn .
∗,
λn
Remark 1.5. It is easy to check that b, μ
S fx ≤ 2 b, μλ fx see, e.g., the proof
of 19 in 15, page 89, we therefore give only the proofs of Theorem 1.2 for b, μΩ and
Theorem 1.3 for b, μ
S .
Remark 1.6. It is easy to see that the condition 1.15 is weaker than Lipβ Sn−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, let β > 0, then for |x| > 2|y|, we have
Ω x − y Ωx C
−
≤ β
γ .
β x − yβ
|x|
|x| log|x|/y
2.1
Now let us return to the proof of Theorem 1.2. Suppose that b, μΩ is a bounded
operator on Lp,ϕ Rn , we are going to prove that b ∈ BMORn .
Journal of Inequalities and Applications
5
We may assume that b, μΩ Lp,ϕ →Lp,ϕ 1. We want to prove that, for any x0 ∈ Rn and
r ∈ R , the inequality
N
1
|Bx0 , r|
holds, where a0 |Bx0 , r|−1
a0 0. Let
Bx0 ,r
b y − a0 dy ≤ A p, Ω, n, γ
2.2
Bx0 ,r
bydy. Since b − a0 , μΩ b, μΩ , we may assume that
f y sgn b y − c0 χBx0 ,r y ,
2.3
where c0 1/|Bx0 , r| Bx0 ,r sgnbydy. Since 1/|Bx0 , r| Bx0 ,r bydy a0 0, we
can easily get |c0 | < 1. Then, f has the following properties:
f ≤ 2,
∞
2.4
2.5
supp f ⊂ Bx0 , r,
f y dy 0,
2.6
Rn
f y b y > 0, y ∈ Bx0 , r,
1
f y b y dy N.
|Bx0 , r| Rn
2.7
2.8
In this proof for j 1, . . . , 15, Aj is a positive constant depending only on Ω, p, n, γ, and
Ai 1 ≤ i < j. Since Ω satisfies 1.2, then there exists an A1 such that 0 < A1 < 1 and
σ
x ∈S
n−1
: Ω x ≥ 2C1
log2/A1 γ
2.9
> 0,
where σ is the measure on Sn−1 which is induced from the Lebesgue measure on Rn . By the
condition 1.15, it is easy to see that
Λ :
x ∈S
n−1
: Ω x ≥ 2C1
log2/A1 2.10
γ
is a closed set. We claim that
if x ∈ Λ and y ∈ Sn−1 , satisfying x − y ≤ A1 , then Ω y ≥ C1
log2/A1 γ .
2.11
6
Journal of Inequalities and Applications
In fact, since |Ωx − Ωy | ≤ C1 /log2/|x − y | γ ≤ C1 /log2/A1 γ , note that Ωx ≥
2C1 /log2/A1 γ , we can get Ωy ≥ C1 /log2/A1 γ . Taking A2 > 3/A1 , let
G x ∈ Rn : |x − x0 | ≥ A2 r, x − x0 ∈ Λ .
2.12
For x ∈ G, we have
b, μΩ fx ≥ μΩ bf x − |bx|μΩ fx
⎧
2 ⎫1/2
⎪
⎪
⎨ ∞ Ω x−y
dt ⎬
b y f y dy 3
t ⎪
⎪
⎩ 0 |x−y|≤t x − yn−1
⎭
⎧
2 ⎫1/2
⎪
⎪
⎨ ∞ Ω x−y
dt ⎬
− |bx|
f
y
dy
t3 ⎪
⎪
⎩ 0 |x−y|≤t x − yn−1
⎭
2.13
: I1 − I2 .
For I1 , noting that if y ∈ Bx0 , r, then |x − x0 | > A2 |y − x0 | for x ∈ G. Thus, we have
y − x0 2
≤
< A1 .
x − y − x − x0 ≤ 2
A2
|x − x0 |
2.14
Using 2.11, we get Ωx − y ≥ C1 /log2/A1 γ . Noting that |x − x0 | |x − y|, it follows
from 2.5, 2.7, 2.8, and Hölder’s inequality that
I1 ≥
⎧
⎪
⎨ ∞
⎪
⎩
⎛
⎜
≥⎝
|x−x0 |
⎛
⎜
⎝
Ω
Bx0 ,r
∞
|x−x0 |
≥
C1
≥
C
Ω
Bx0 ,r
log2/A1 log2/A1 ⎞2 ⎫1/2
⎪
x−y b y f y
⎟ dt ⎬
χ{|x−y|≤t} y dy⎠ 3
t ⎪
x − yn−1
⎭
⎞
−1/2
∞
x−y b y f y
dt
dt ⎟
χ{|x−y|≤t} dy 3 ⎠
3
t
x − yn−1
|x−x0 | t
γ |x − x0 |
γ |x − x0 |
A3 Nr n |x − x0 |−n .
x − y−n1 b y f y
Bx0 ,r
−n
b y f y dy
Bx0 ,r
|x−x0 |≤t
|x−y|≤t
dt
dy
t3
2.15
Journal of Inequalities and Applications
7
For x ∈ G, by Ω ∈ L∞ Sn−1 , 2.4, 2.5, 2.6, the Minkowski inequality, and Lemma 2.1, we
obtain
⎧
2 ⎫1/2
⎨ ∞ dt ⎬
Ω x−y
Ωx − x0 dy
f y
χ
−
χ
I2 |bx|
3
{|x−y|≤t}
{|x−x
|≤t}
0
n−1
n−1
t ⎭
⎩ 0 Rn
x − y
|x − x0 |
⎧⎛
⎪
⎨ ∞ ⎝
≤ |bx|
⎪
⎩
0
|x−y|≤t<|x−x0 |
⎛
⎝
∞ |x−x0 |≤t<|x−y|
0
⎛
⎜
⎝
∞
0
≤ |bx|
|Ωx − x0 | f y dy
|x − x0 |n−1
2
⎞1/2
dt ⎠
t3
⎫
⎞2 ⎞1/2 ⎪
⎪
Ω x − y Ωx − x dt ⎟ ⎬
0 ⎠
⎝
dy
f
y
−
⎠
|x−x0 |≤t ⎪
t3
x − yn−1 |x − x0 |n−1 ⎪
⎭
|x−y|≤t
⎛
⎧
⎨
⎩
2 ⎞1/2
Ω x − y dt ⎠
n−1 f y dy
t3
x − y
1/2
Ω x − y dt
f y
dy
3
x − yn−1
|x−y|≤t<|x−x0 | t
Bx0 ,r
Bx0 ,r
|Ωx − x0 | f y
|x − x0 |n−1
|x−y|>t≥|x−x0 |
dt
t3
1/2
dy
⎞1/2 ⎫
⎛
⎪
⎬
Ω x − y
dt ⎠
Ωx − x0 ⎝
f y −
dy
3
n−1
n−1
|x−y|≤t t
x − y
⎪
|x − x0 | ⎭
Bx0 ,r
|x−x |≤t
0
≤ C|bx| r
1/2
Bx0 ,r
f y |x − x0 |n1/2
f y dy Bx0 ,r
γ dy
|x − x0 |n log|x − x0 |/r
|x − x0 | −γ
.
≤ A4 |bx|r n |x − x0 |−n log
r
2.16
Let
F
!
"
A3 N
|x − x0 | γ
x ∈ G : |bx| >
, |x − x0 | < N 1/n r .
log
2A4
r
2.17
Without loss of generality, we may assume that N > A2 > 1, otherwise, we get the desired
8
Journal of Inequalities and Applications
result. Since ϕt is nonincreasing, it follows that ϕ|Bx0 , N 1/n r| ≤ ϕ|Bx0 , r| ϕr n . By
2.13, 2.15, and 2.16, we have
p
f p,ϕ ≥ b, μΩ f p p,ϕ
L
L
1
b, μΩ fxp dx
≥ p
1/n
1/n
B x0 , N r
ϕ B x0 , N r
|x−x0 |<N 1/n r
p
1
1
−n
n
A
≥
Nr
dx
−
x
|x
|
3
0
p
ϕr n Nr n G\F∩{|x−x0 |<N 1/n r} 2
p
1
1
−n
n
A
≥
Nr
dx
−
x
|x
|
p
3
0
ϕr n Nr n {A5 |F|A2 rn 1/n <|x−x0 |<N 1/n r}∩G 2
ωn−1
p
ϕr n Nr n
ωn−1
≥
ϕr n p Nr n A3 Nr n
2
p N 1/n r
A5 |F|A2 r
p−1 A3 /2
p
n − np
n 1/n
2.18
t−pnn−1 dt
1−pn N 1−p r n1−p − A5
|F| A2 rn
1−p .
Thus,
|F| A2 rn
1−p
p ≤ A6 N 1−p r n1−p 1 ϕr n p f Lp,ϕ .
2.19
Now, we claim that
f p,ϕ ≤
L
C
,
ϕr n 2.20
where C is independent of r. In fact,
f p,ϕ sup
L
1
n ϕ|Bx, t|
x∈R
t>0
1
|Bx, t|
p
f y dy
Bx,t
1/p
.
2.21
Now, we consider the Lp,ϕ norm of f in the following two cases.
Case 1 t > r. Since s1/p ϕs is nondecreasing in s, then
1
1
1
1
≤
.
n
1/p
n/p
ϕ|Bx, t| |Bx, t|
ϕr r
2.22
Journal of Inequalities and Applications
9
Thus,
f Lp,ϕ
1
1
≤ sup
n
n/p
x∈Rn ϕr r
t>0
≤
1/p
Bx,t
t>0
1
1
sup
n
n/p
x∈Rn ϕr r
p
f y dy
p
f y dy
1/p
2.23
Bx,t∩Bx0 ,r
C
.
ϕr n Case 2 t ≤ r. Since ϕs is nonincreasing in s, then
1
1
≤
.
ϕ|Bx, t| ϕr n 2.24
Thus,
f p,ϕ ≤ sup
L
1
n
x∈Rn ϕr t>0
≤
1
|Bx, t|
p
f y dy
1/p
Bx,t
2.25
C
.
ϕr n Now, 2.20 is established. Then, by 2.19 and 2.20, we get
|F| A2 rn ≥ A7 Nr n .
2.26
−1 n
n
If N ≤ 2A−1
7 A2 , then Theorem 1.2 is proved. If N > 2A7 A2 , then
|F| ≥
A7
Nr n .
2
2.27
Let gy χBx0 ,r y. For x ∈ F, we have
⎧
2 ⎫1/2
⎪
⎪
∞ ⎨
Ω
x
−
y
dt ⎬
b, μΩ gx ≥ |bx|
g y dy 3
t ⎪
⎪
⎩ 0 |x−y|≤t x − yn−1
⎭
⎧
2 ⎫1/2
⎪
⎬
⎨ ∞ dt ⎪
Ω x−y
−
b
y
g
y
dy
t3 ⎪
⎪
⎭
⎩ 0 |x−y|≤t x − yn−1
: K1 − K2 .
2.28
10
Journal of Inequalities and Applications
Noting that if y ∈ Bx0 , r and x ∈ F, we get |x − y − x − x0 | ≤ A1 . Applying 2.11, we
have Ωx − y ≥ C1 /log2/A1 γ . Since |x − y| |x − x0 | when y ∈ Bx0 , r and x ∈ F, it
follows that
⎧
⎛
⎞2 ⎫1/2
⎪
⎪
⎨ ∞
Ω x−y
⎟ dt ⎬
⎜
K1 ≥ |bx|
⎝
n−1 χ{|x−y|≤t} y dy⎠ 3 ⎪
⎪
t ⎭
⎩ |x−x0 |
Bx0 ,r x − y ⎛
⎜
≥ |bx|⎝
∞
|x−x0 |
C1 |bx|
≥
log2/A1 Bx0 ,r
⎞
−1/2
∞
Ω x−y
dt
dt ⎟
χ{|x−y|≤t} dy 3 ⎠
3
t
x − yn−1
|x−x0 | t
γ |x − x0 |
≥ A8 |bx||x − x0 |−n
x − y−n1
2.29
Bx0 ,r
|x−x0 |≤t
|x−y|≤t
dt
dy
t3
dy
Bx0 ,r
A8 r n |bx||x − x0 |−n .
By Ω ∈ L∞ Sn−1 , |x − x0 | |x − y| when y ∈ Bx0 , r and x ∈ F and the Minkowski inequality,
we have
K2 ≤ C
Bx0 ,r
b y dy
x − yn
≤ A9 |x − x0 |
−n
b y dy
2.30
Bx0 ,r
A9 Nr n |x − x0 |−n .
Thus, by 2.28, 2.29, and 2.30, we get, for x ∈ F,
b, μΩ gx ≥ A8 r n |bx||x − x0 |−n − A9 Nr n |x − x0 |−n .
2.31
Similar to the proof of 2.20, we can easily get gLp,ϕ ≤ C/ϕr n . Thus, by 2.31,
Journal of Inequalities and Applications
11
ϕNr n ≤ ϕr n , and |bx| > NA3 /2A4 log|x − x0 |/rγ when x ∈ F, we have
A10
≥ g Lp,ϕ ≥ b, μΩ g Lp,ϕ
ϕr n 1/p
p
1
≥
b, μΩ gx dx
ϕNr n Nr n 1/p
|x−x0 |<N 1/n r
−1/p
1
≥
dx
b, μΩ gx dx
ϕr n Nr n 1/p |x−x0 |<N 1/n r
|x−x0 |<N 1/n r
1
b, μΩ gxdx
≥
ϕr n Nr n F
A8 r n
A9 Nr n
−n
≥
|bx||x − x0 | dx −
|x − x0 |−n dx
ϕr n Nr n F
ϕr n Nr n F
A11
|x − x0 | γ
≥
log
|x − x0 |−n dx
ϕr n F
r
A9
−
|x − x0 |−n dx
ϕr n F
2.32
: L1 − L2 .
We first estimate L2 . Since A2 r < |x − x0 | < N 1/n r for x ∈ F, we have
A9 ωn−1
L2 ≤
ϕr n N 1/n r
A2 r
ρ−1 dρ ≤
A12
log N.
ϕr n 2.33
Now, the estimate of L1 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 ≥ 3 and 3r < A2 r < |x − x0 | <
N 1/n r for x ∈ F, by 2.27, we get
log|x − x0 |/r n
|x − x0 | γ−n
dx
log
r
|x − x0 |/r
F
n
γ−n
log N 1/n
A7 A11 N
≥
log A2
2ϕr n N 1/n
A11 r −n
L1 ϕr n ≥
n
A13 log N .
ϕr n 2.34
12
Journal of Inequalities and Applications
Case 2 1 < γ < n. Since the function log sγ /sn is decreasing for s ≥ 3 and 3r < A2 r <
|x − x0 | < N 1/n r for x ∈ F, by 2.27, we have
log|x − x0 |/r
|x − x0 |/rn
F
γ
log N 1/n
A7 A11
≥
N
2ϕr n N
A11 r −n
L1 ϕr n ≥
γ
dx
2.35
γ
A14 log N .
n
ϕr From Cases 1 and 2, we know that there exists a constant τ > 1 such that
L1 ≥
τ
A15 log N .
ϕr n 2.36
So by 2.32, 2.33, and 2.36, we get
τ
A10 ≥ A15 log N − A12 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.
p,ϕ
n
Suppose that b, μ
S is a bounded operator on L R , we are going to prove that
n
b ∈ BMOR .
n
We may assume that b, μ
S Lp,ϕ →Lp,ϕ 1. We want to prove that, for any x0 ∈ R and
r ∈ R , the inequality
1
N
|Bx0 , r|
b y − a0 dy ≤ B Ω, p, n, Bx0 ,r
3.1
holds, where a0 |Bx0 , r|−1 Bx0 ,r bydy. Since b − a0 , μ
S b, μS , we may assume that
a0 0. Let fy be as 2.3, then 2.4–2.8 hold. In this proof for j 1, . . . , 13, Bj is a positive
constant depending only on Ω, p, n, , and Bi 1 ≤ i < j. Since Ω satisfies 1.2, then there
exists a B1 such that 0 < B1 < 1 and
σ
x ∈S
n−1
: Ω x ≥ 2C1
log2/B1 γ
> 0,
3.2
Journal of Inequalities and Applications
13
where σ is the measure on Sn−1 which is induced from the Lebesgue measure on Rn . By the
condition 1.15, it is easy to see that
Λ :
x ∈S
n−1
: Ω x ≥ 2C1
log2/B1 3.3
γ
is a closed set. As the proof of 2.11, we can get the following:
if x ∈ Λ and y ∈ Sn−1 , satisfying x − y ≤ B1 , then Ω y ≥ C1
log2/B1 γ .
3.4
Taking B2 > 3/B1 1, let
G x ∈ Rn : |x − x0 | ≥ B2 r, x − x0 ∈ Λ .
3.5
For x ∈ G, we have
⎛
∞ b, μ fx ⎝
S
0
⎛
≥⎝
⎞1/2
2
dydt
Ω
y
−
z
n− bx − bzfzdz n12 ⎠
t
|x−y|<t |y−z|<t y − z
∞
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
×
|y−z|<t
⎛
≥⎝
∞
⎞1/2
2
dydt
Ω y−z
bx − bzfzdz n12 ⎠
t
y − zn−
4|x−x0 |
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |, y−x0 ∈Λ
⎞1/2
2
dydt
Ω
y
−
z
×
bzfzdz n12 ⎠
|y−z|<t y − zn−
t
⎛
− |bx|⎝
∞
4|x−x0 |
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
|y−z|<t
⎞1/2
2
dydt
Ω y−z
fzdz n12 ⎠
t
y − zn−
: I1 − I2 .
3.6
14
Journal of Inequalities and Applications
For I1 , noting that if |z − x0 | < r, |x − x0 | > B2 |z − x0 |, and |y − x0 | > 2B2 |z − x0 |, then we get
1
|z − x0 |
≤
< B1 .
y − z − y − x0 ≤ 2 y − x0 B2
3.7
Then by 3.4, we get Ωy − z ≥ C1 /log2/B1 γ . Since 4|x−x0 | > |y−x0 ||z−x0 | ≥ |y−z| ≥
|y−x0 |−|z−x0 | > 2|x−x0 |−|x−x0 |/2 3|x−x0 |/2 and 4|x−x0 | > |x−y| ≥ |y−x0 |−|x−x0 | > |x−x0 |,
we get 4|x − x0 | ≥ |y − z| ≥ 3|x − x0 |/2 and 4|x − x0 | > |x − y| > |x − x0 |. Thus, by 2.5, 2.7,
2.8, and the Hölder inequality, we get
I1 ≥ C
∞
|x−y|<t, y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
⎛
×⎝
∞
Bx0 ,r
|x−y|<t, y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
2−n
tn12
bzfz
bzfz
Bx0 ,r
dtdy
4|x−x0 |<t, |x−y|<t
|y−z|<t
C|x − x0 |2−n
≥ C|x − x0 |
y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
Bx0 ,r
−n
⎞−1/2
dydt ⎠
≥ C|x − x0 |
Ω y−z
dydt
bzfzχ{|y−z|<t} dz n12
y − zn−
t
y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |<t
tn12
dz
3.8
dtdy
dz
tn12
bzfzdz
Bx0 ,r
B3 Nr n |x − x0 |−n .
By 2.5 and 2.6, we have
⎛
I2 |bx|⎝
∞
4|x−x0 |
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
⎞1/2
2
Ω
y
−
x
Ω
y
−
z
dydt
0
fzdz n12 ⎠
×
χ
−
χ
t
Rn y − zn− {|y−z|<t} y − x0 n− {|y−x0 |<t}
Journal of Inequalities and Applications
⎛
∞
≤ |bx|⎝
4|x−x0 |
15
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
⎞1/2
2
Ω y − x0
Ω y−z
dydt ⎟
× |y−z|<t n− − n− fzdz n12 ⎠
t
y−z
y − x0
|y−x |<t
⎛
⎜
|bx|⎝
⎛
⎜
|bx|⎝
0
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
⎞1/2
2
dydt ⎟
Ω y−z
n− fzdz n12 ⎠
|y−z|<t |y−x |≥t y − z
t
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
|y−z|≥t
|y−x |<t
∞
4|x−x0 |
∞
4|x−x0 |
0
⎞1/2
2
dydt ⎟
Ω y − x0
fzdz n12 ⎠
y − x0 n−
t
0
: I21 I22 I23 .
3.9
In I22 , we have t ≤ |y − x0 | < 3|x − x0 | and t ≥ 4|x − x0 |. In I23 , we get t ≤ |y − z| < 4|x − x0 |
and t ≥ 4|x − x0 |. It is easy to see that I22 I23 0. Now, we estimate I21 , by Ω ∈ L∞ Sn−1 , the
Minkowski inequality, Lemma 2.1 for |y − x0 | > 2|z − x0 |, and 2.4, we get
⎛
fzdz⎝
I21
≤ C|bx|
Bx0 ,r
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |≤t, |y−z|<t
|y−x0 |<t, |x−y|≤t
1
dtdy
1/2
×
y − x0 2n− logy − x0 /r 2γ tn12
3.10
|x − x0 | −γ
≤ B4 |bx|r n |x − x0 |−n log
.
r
From 3.9 and 3.10, we get
|x − x0 | −γ
I2 ≤ B4 |bx|r n |x − x0 |−n log
.
r
3.11
Let
!
F
"
B3 N
|x − x0 | γ
1/n
x ∈ G : |bx| >
, |x − x0 | < N r .
log
2B4
r
3.12
16
Journal of Inequalities and Applications
Without loss of generality, we may assume that N > B2 > 1, otherwise, we get the desired
result. Since ϕt is nonincreasing, we have ϕ|Bx0 , N 1/n r| ≤ ϕ|Bx0 , r| ϕr n . Then by,
3.6, 3.8, and 3.11, we get
p
f p,ϕ ≥ b, μ f p p,ϕ
S
L
L
1
≥ ϕ B x0 , N 1/n r p B x0 , N 1/n r 1
≥
p
n
ϕr Nr n
1
p
n
ϕr Nr n
≥
1
p
n
ϕr Nr n
1
≥
ϕr n G\F∩{|x−x0 |<N 1/n r}
|x−x0 |<N 1/n r
B3 Nr n
2
p σΛNr n 1/n
<|x−x0 |<N 1/n r}∩G
p N 1/n r
B5 |F|B2 r
p−1 B3 /2
n 1/n
t
p
dx
1
B3 Nr n |x − x0 |−n
2
−pnn−1
p
n − np
S
1
B3 Nr n |x − x0 |−n
2
{B5 |F|B2 rn b, μ fxp dx
dt
Λ
dx
J x dσ x
1−pn N 1−p r n1−p − B5
p
|F| B2 rn
1−p .
3.13
Thus,
|F| B2 rn
1−p
p p ≤ B6 N 1−p r n1−p 1 ϕr n f Lp,λ .
3.14
|F| B2 rn ≥ B7 Nr n .
3.15
Then, by 2.20 and 3.14, we get
If N ≤ 2B7−1 B2n , then Theorem 1.3 is proved. If N > 2B7−1 B2n , then
|F| ≥
B7
Nr n .
2
3.16
Journal of Inequalities and Applications
17
Let gy χBx0 ,r y. For x ∈ F, we have
⎛
∞ b, μ gx ⎝
S
0
⎞1/2
2
dydt
Ω
y
−
z
n− bx − bzgzdz n12 ⎠
t
|x−y|<t |y−z|<t y − z
⎛
∞
≥⎝
⎞1/2
2
dydt
Ω
y−z
bx−bzgzdz n12 ⎠
t
|y−z|<t y−zn−
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
⎛
≥ |bx|⎝
∞
4|x−x0 |
⎛
∞
⎝
−
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |, y−x0 ∈Λ
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
|y−z|<t
|y−z|<t
⎞1/2
2
dydt
Ω y−z
gzdz n12 ⎠
t
y−zn−
⎞1/2
2
dydt
Ω y−z
bzgzdz n12 ⎠
t
y − zn−
: K1 − K2 .
3.17
For K1 , as above mentioned, we have Ωy − z ≥ C1 /log2/B1 γ . Since 4|x − x0 | ≥ |y − z| ≥
3|x − x0 |/2 and 4|x − x0 | > |x − y| > |x − x0 |, it follows the Hölder inequality that
K1 |bx|
⎧
⎨ ∞
⎩
≥ |bx|
⎛
×⎝
|x−y|<t, y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
∞
4|x−x0 |
|x−y|<t, y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
≥ C|bx||x − x0 |
≥ B8 N|x − x0 |
Bx0 ,r
⎫1/2
2
Ω y−z
dydt ⎬
χ
dz
y−zn− {|y−z|<t}
tn12 ⎭
Ω y−z
dydt
χ
dz
y − zn− {|y−z|<t} tn12
⎞−1/2
dydt ⎠
tn12
2−n
Bx0 ,r
−n
Bx0 ,r
|x−y|<t, y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
∞
y−x0 ∈Λ
2|x−x0 |<|y−x0 |<3|x−x0 |
dtdy
4|x−x0 |<t
tn12
dz
dz
Bx0 ,r
B8 Nr n |x − x0 |−n .
3.18
18
Journal of Inequalities and Applications
By Ω ∈ L∞ Sn−1 , the Minkowski inequality, and |x−x0 | |y−z| for 2|x−x0 | < |y−x0 | < 3|x−x0 |
and z ∈ Bx0 , r, we get
⎛
∞
K2 ⎝
|x−y|<t
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
≤ B9 |x − x0 |
C
≤
|x − x0 |n−
|bz|dz
Bx0 ,r
−n
⎞1/2
2
dydt
Ω y−z
bzχ{|y−z|<t} dz n12 ⎠
t
Bx0 ,r y − zn−
∞
2|x−x0 |<|y−x0 |<3|x−x0 |
4|x−x0 |
dtdy
1/2
tn12
|bz|dz
Bx0 ,r
B9 Nr n |x − x0 |−n .
3.19
Thus, by 3.17, 3.18, and 3.19, we get, for x ∈ F,
b, μ gx ≥ B8 r n |bx||x − x0 |−n − B9 Nr n |x − x0 |−n .
S
3.20
Thus, by 3.20, ϕNr n ≤ ϕr n , |bx| > NB3 /2B4 log|x − x0 |/rγ when x ∈ F and the
Hölder inequality, we have
B10
≥ g Lp,ϕ ≥ b, μ
S g Lp,ϕ
n
ϕr 1/p
p
1
≥
b, μS gx dx
ϕNr n Nr n 1/p
|x−x0 |<N 1/n r
−1/p
1
≥
dx
b, μS gx dx
ϕr n Nr n 1/p |x−x0 |<N 1/n r
|x−x0 |<N 1/n r
1
b, μ gxdx
≥
S
ϕr n Nr n F
B8 r n
B9 Nr n
−n
≥
|bx||x − x0 | dx −
|x − x0 |−n dx
ϕr n Nr n F
ϕr n Nr n F
B11
|x − x0 | γ
≥
log
|x − x0 |−n dx
ϕr n F
r
B9
−
|x − x0 |−n dx
ϕr n F
: L1 − L2 .
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
L1 ≥
τ
B12 log N ,
n
ϕr B13
L2 ≤
log N.
ϕr n 3.22
So, by 3.21 and 3.22, we get
τ
B10 ≥ B12 log N − B13 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,
SRFDP of China Grant no.: 20090003110018, and NSF of Zhenjiang Grant no.: Y7080325.
References
1 E. M. Stein, “On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz,” Transactions of the
American Mathematical Society, vol. 88, pp. 430–466, 1958.
2 Y. Ding, D. Fan, and Y. Pan, “Weighted boundedness for a class of rough Marcinkiewicz integrals,”
Indiana University Mathematics Journal, vol. 48, no. 3, pp. 1037–1055, 1999.
3 S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff, “Some weighted norm inequalities concerning the
Schrödinger operators,” Commentarii Mathematici Helvetici, vol. 60, no. 2, pp. 217–246, 1985.
4 C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, vol. 83
of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Washington, DC,
USA, 1994.
5 Q. Xue and Y. Ding, “Weighted Lp boundedness for parametrized Littlewood-Paley operators,”
Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1143–1165, 2007.
6 M. Sakamoto and K. Yabuta, “Boundedness of Marcinkiewicz functions,” Studia Mathematica, vol. 135,
no. 2, pp. 103–142, 1999.
7 A. Torchinsky and S. L. Wang, “A note on the Marcinkiewicz integral,” Colloquium Mathematicum, vol.
60-61, no. 1, pp. 235–243, 1990.
8 E. M. Stein, “The development of square functions in the work of A. Zygmund,” Bulletin of the
American Mathematical Society, vol. 7, no. 2, pp. 359–376, 1982.
9 Y. Ding, S. Lu, and K. Yabuta, “On commutators of Marcinkiewicz integrals with rough kernel,”
Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 60–68, 2002.
10 Y. Ding and Q. Xue, “Endpoint estimates for commutators of a class of Littlewood-Paley operators,”
Hokkaido Mathematical Journal, vol. 36, no. 2, pp. 245–282, 2007.
11 R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several
variables,” The Annals of Mathematics, vol. 103, no. 3, pp. 611–635, 1976.
12 S. Chanillo, “A note on commutators,” Indiana University Mathematics Journal, vol. 31, no. 1, pp. 7–16,
1982.
13 T. Mizuhara, “Commutators of singular integral operators on Morrey spaces with general growth
functions,” Sūrikaisekikenkyūsho Kōkyūroku, no. 1102, pp. 49–63, 1999, Proceedings of the Coference on
Harmonic Analysis and Nonlinear Partial Differential Equations, Kyoto, Japan, 1998.
14 Y. Komori and T. Mizuhara, “Factorization of functions in H 1 Rn and generalized Morrey spaces,”
Mathematische Nachrichten, vol. 279, no. 5-6, pp. 619–624, 2006.
20
Journal of Inequalities and Applications
15 E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series,
no. 30, Princeton University Press, Princeton, NJ, USA, 1970.
16 A. Uchiyama, “On the compactness of operators of Hankel type,” Tôhoku Mathematical Journal, vol.
30, no. 1, pp. 163–171, 1978.
17 Y. Ding, “A note on end properties of Marcinkiewicz integral,” Journal of the Korean Mathematical
Society, vol. 42, no. 5, pp. 1087–1100, 2005.