Journal of Inequalities and Applications Volume 2010, Article ID 961502,20pages doi:10.1155/2010/961502
Research Article
Commutators of Littlewood-Paley Operators on the
Generalized Morrey Space
Yanping Chen,
1Yong Ding,
2and Xinxia Wang
31Department of Mathematics and Mechanics, Applied Science School, University of Science and Technology
Beijing, Beijing 100083, China
2Laboratory of Mathematics and Complex Systems (BNU), School of Mathematical Sciences, Beijing
Normal University, Ministry of Education, Beijing 100875, China
3The College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang 830046, China
Correspondence should be addressed to Yanping Chen,[email protected]
Received 6 May 2010; Accepted 11 July 2010
Academic Editor: Shusen Ding
Copyrightq2010 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.
LetμΩ,μS, and μ∗ ,
λ denote the Marcinkiewicz integral, the parameterized area integral, and
the parameterized Littlewood-Paleyg∗λ function, respectively. In this paper, the authors give a characterization of BMO space by the boundedness of the commutators ofμΩ,μS, andμ
∗, λ on
the generalized Morrey spaceLp,ϕRn.
1. Introduction
LetSn−1 {x∈Rn : |x| 1}be the unit sphere inRn equipped with the Lebesgue measure dσ. Suppose thatΩsatisfies the following conditions.
a Ωis the homogeneous function of degree zero onRn\ {0}, that is,
Ωμx Ωx, for anyμ >0, x∈Rn\ {0}. 1.1
b Ωhas mean zero onSn−1, that is,
Sn−1
c Ω∈LipSn−1, that is,
Ω
x−Ωy≤x−y, for any x, y∈Sn−1. 1.3
In 1958, Stein1defined the Marcinkiewicz integral of higher dimensionμΩas
μΩfx
∞
0
|FΩ,tx|2 dt
t3 1/2
, 1.4
where
FΩ,tx
|x−y|≤t
Ωx−y x−yn−1 f
ydy. 1.5
We refer to see1,2for the properties ofμΩ.
Let 0< < nandλ >1.The parameterized area integralμS and the parameterized Littlewood-Paleygλ∗functionμ∗λ, are defined by
μ Sfx
⎛
⎝
Γx
1
t
|y−z|<t
Ωy−z
y−zn−fzdz 2
dydt tn 1
⎞ ⎠
1/2
, 1.6
whereΓx {y, t∈Rn 1 :|x−y|< t},and
μ∗λ,fx ⎛
⎝
Rn1
t t x−y
λn
1
t
|y−z|<t
Ωy−z
y−zn−fzdz 2
dydt tn 1
⎞ ⎠
1/2
, 1.7
respectively.μS andμ∗λ, play very important roles in harmonic analysis and PDEe.g., see 3–8.
Before stating our result, let us recall some definitions. For b ∈ LlocRn, the
commutatorb, μΩformed byband the Marcinkiewicz integralμΩare defined by
b, μΩfx ⎛ ⎝∞
0
|x−y|≤t
Ωx−y x−yn−1
bx−byfydy 2
dt t3
⎞ ⎠
1/2
Let 0< < nandλ >1.The commutatorb, μSofμS and the commutatorb, μ∗λ,ofμ∗λ,
are defined, respectively, by
b, μSfx ⎛
⎝
Γx
1
t
|y−z|≤t
Ωy−z
y−zn−bx−bzfzdz 2
dydt tn 1
⎞ ⎠
1/2
, 1.9
b, μ∗λ,fx ⎛
⎝
Rn1
t t x−y
λn
× 1
t
|y−z|≤t
Ωy−z
y−zn−bx−bzfzdz 2
dydt tn 1
⎞ ⎠
1/2
.
1.10
Letb∈LlocRn. It is said thatb∈BMORnif
b∗:sup
B⊂Rn
Mb, B<∞, 1.11
whereBBx, rdenotes the ball inRncentered atxand with radiusr,
Mb, B 1
|B|
B
|bx−bB|dx, 1.12
andbB 1/|B|
Bbydy.
There are some results about the boundedness of the commutators formed by BMO functions withμΩ,μ
S, andμ
∗,
λ see7,9,10.
Many important operators gave a characterization of BMO space. In 1976, Coifman et al.11gave a characterization of BMO space by the commutator of Riesz transform; in 1982, Chanillo12studied 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 μΩ, μS, and μ
∗,
λ on the generalized Morrey space Lp,ϕRn.
Definition 1.1. Let 1 < p < ∞. Suppose that ϕ : 0,∞ → 0,∞ be such that ϕt is nonincreasing andt1/pϕtis nondecreasing. The generalized Morrey spaceLp,ϕ is defined by
Lp,ϕRn f∈LlocRn:fLp,ϕ<∞
, 1.13
where
f
Lp,ϕ sup
x∈Rn
r>0
1
ϕ|Bx, r|
1
|Bx, r|
Bx,r f
ypdy 1/p
We refer to see13,14 for the known results of the generalized Morrey spaceLp,ϕ
for some suitableϕ. Noting thatϕt ≡t−1/p, we get the Lebesque spaceLpRn. Forϕt tλ/n−1/p 0< λ < n,Lp,ϕRncoincides with the Morrey spaceLp,λRn.
The main result in this paper is as follows.
Theorem 1.2. Assume thatϕtis nonincreasing andt1/pϕtis nondecreasing. Suppose thatb, μ Ω is defined as1.8,Ωsatisfies1.1,1.2, and
Ω
x−Ωy≤ C1
log2/x−yγ, C1>0, γ >1, x
, y∈Sn−1. 1.15
Ifb, μΩis bounded onLp,ϕRnfor somep1< p <∞, thenb∈BMORn.
Theorem 1.3. Let0 < < nand1 < p < ∞. Assume thatϕtis nonincreasing andt1/pϕtis nondecreasing. Suppose thatb, μ
Sis defined as1.9,Ωsatisfies1.1,1.2, and1.15. Ifb, μS is a bounded operator onLp,ϕRnfor somep 1< p <∞, thenb∈BMORn.
Theorem 1.4. Let0 < < n, λ > 1, and 1 < p < ∞. Assume thatϕt is nonincreasing and t1/pϕt is nondecreasing. Suppose that b, μ∗,ϕ
λ is defined as1.10, Ω satisfies1.1,1.2, and
1.15. Ifb, μ∗λ,is onLp,ϕRnfor somep1< p <∞, thenb∈BMORn.
Remark 1.5. It is easy to check that b, μSfx ≤ 2λnb, μ∗λ,fx see, e.g., the proof
of 19 in15, page 89, we therefore give only the proofs ofTheorem 1.2for b, μΩand
Theorem 1.3forb, μ S.
Remark 1.6. It is easy to see that the condition1.15is weaker than LipβSn−1for 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 of17, we can easily get the following.
Lemma 2.1. IfΩsatisfies conditions1.1,1.2, and1.15, letβ >0,then for|x|>2|y|, we have
Ωx−y x−yβ −
Ωx
|x|β ≤
C
|x|βlog|x|/yγ. 2.1
Now let us return to the proof of Theorem 1.2. Suppose that b, μΩ is a bounded
We may assume thatb, μΩLp,ϕ→Lp,ϕ 1. We want to prove that, for anyx0∈Rnand r∈R , the inequality
N 1
|Bx0, r|
Bx0,r
by−a0dy≤A
p,Ω, n, γ 2.2
holds, wherea0|Bx0, r|−1
Bx0,rbydy.Sinceb−a0, μΩ b, μΩ,we may assume that
a00.Let
fysgnby−c0
χBx0,r
y, 2.3
wherec0 1/|Bx0, r|
Bx0,rsgnbydy.Since1/|Bx0, r|
Bx0,rbydy a0 0, we
can easily get|c0|<1.Then,fhas the following properties:
f
∞≤2, 2.4
suppf⊂Bx0, r, 2.5
Rn
fydy0, 2.6
fyby>0, y∈Bx0, r, 2.7
1
|Bx0, r|
Rn
fybydyN. 2.8
In this proof forj 1, . . . ,15,Aj is a positive constant depending only onΩ, p,n,γ, and
Ai 1≤i < j. SinceΩsatisfies1.2, then there exists anA1such that 0< A1<1 and
σ
x∈Sn−1: Ωx≥ 2C1
log2/A1 γ
>0, 2.9
whereσis the measure onSn−1which is induced from the Lebesgue measure onRn. By the
condition1.15, it is easy to see that
Λ:
x∈Sn−1: Ωx≥ 2C1
log2/A1 γ
2.10
is a closed set. We claim that
ifx∈Λ andy∈Sn−1, satisfyingx− y ≤A
1, then Ω
y≥ C1
log2/A1
In fact, since|Ωx−Ωy| ≤ C1/log2/|x−y|γ ≤ C1/log2/A1γ,note thatΩx ≥
2C1/log2/A1γ,we can getΩy≥C1/log2/A1γ.TakingA2>3/A1, let
Gx∈Rn:|x−x0| ≥A2r, x−x0∈Λ
. 2.12
Forx∈G, we have
b, μΩ
fx≥μΩbfx− |bx|μΩfx
⎧ ⎪ ⎨ ⎪ ⎩ ∞ 0
|x−y|≤t
Ωx−y x−yn−1 b
yfydy 2 dt t3 ⎫ ⎪ ⎬ ⎪ ⎭
1/2
− |bx| ⎧ ⎪ ⎨ ⎪ ⎩ ∞ 0
|x−y|≤t
Ωx−y x−yn−1 f
ydy
2 dt t3 ⎫ ⎪ ⎬ ⎪ ⎭
1/2
:I1−I2.
2.13
ForI1,noting that ify∈Bx0, r, then|x−x0|> A2|y−x0|forx∈G. Thus, we have
x−y−x−x0≤2
y−x0
|x−x0| ≤
2
A2
< A1. 2.14
Using2.11, we getΩx−y ≥C1/log2/A1γ.Noting that|x−x0| |x−y|,it follows
from2.5,2.7,2.8, and H ¨older’s inequality that
I1≥ ⎧ ⎪ ⎨ ⎪ ⎩ ∞
|x−x0| ⎛ ⎜ ⎝
Bx0,r
Ωx−ybyfy
x−yn−1 χ{|x−y|≤t}
ydy ⎞ ⎟ ⎠ 2 dt t3 ⎫ ⎪ ⎬ ⎪ ⎭
1/2
≥
⎛ ⎜ ⎝
∞
|x−x0|
Bx0,r
Ωx−ybyfy
x−yn−1 χ{|x−y|≤t}dy dt t3 ⎞ ⎟ ⎠ ∞
|x−x0|
dt t3
−1/2
≥ C1
log2/A1
γ|x−x0|
Bx0,r
x−y−n 1byfy
|x−x0|≤t
|x−y|≤t dt t3dy
≥ C
log2/A1
γ|x−x0|−n
Bx0,r
byfydy
A3Nrn|x−x0|−n.
Forx∈G, byΩ∈L∞Sn−1,2.4,2.5,2.6, the Minkowski inequality, andLemma 2.1, we
obtain
I2|bx| ⎧ ⎨ ⎩ ∞ 0 Rn
fy
Ωx−y
x−yn−1χ{|x−y|≤t}−
Ωx−x0
|x−x0|n−1
χ{|x−x0|≤t} dy 2 dt t3 ⎫ ⎬ ⎭
1/2
≤ |bx| ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎝∞ 0
|x−y|≤t<|x−x0| Ω
x−y x−yn−1
f ydy
2 dt t3
⎞ ⎠
1/2
⎛ ⎝∞
0
|x−x0|≤t<|x−y|
|Ωx−x0|
|x−x0|n−1
fydy 2
dt t3
⎞ ⎠
1/2
⎛ ⎜ ⎝ ∞ 0 ⎛ ⎝
|x−x0|≤t
|x−y|≤t
Ωx−y x−yn−1 −
Ωx−x0
|x−x0|n−1 f
ydy
⎞ ⎠ 2 dt t3 ⎞ ⎟ ⎠
1/2⎫⎪⎪ ⎬ ⎪ ⎪ ⎭
≤ |bx| ⎧ ⎨ ⎩
Bx0,r Ω
x−y x−yn−1
f y
|x−y|≤t<|x−x0|
dt t3
1/2 dy
Bx0,r
|Ωx−x0|
|x−x0|n−1 f
y
|x−y|>t≥|x−x0|
dt t3
1/2 dy
Bx0,r
fyΩ
x−y x−yn−1 −
Ωx−x0
|x−x0|n−1
⎛ ⎝
|x−y|≤t
|x−x0|≤t
dt t3
⎞ ⎠
1/2
dy ⎫ ⎪ ⎬ ⎪ ⎭
≤C|bx|
r1/2
Bx0,r f
y
|x−x0|n 1/2 dy
Bx0,r
f y
|x−x0|n
log|x−x0|/r γdy
≤A4|bx|rn|x−x0|−n
log|x−x0|
r −γ . 2.16 Let F !
x∈G: |bx|> A3N
2A4
log|x−x0|
r γ
, |x−x0|< N1/nr "
. 2.17
result. Sinceϕtis nonincreasing, it follows thatϕ|Bx0, N1/nr|≤ϕ|Bx0, r| ϕrn. By
2.13,2.15, and2.16, we have
fpLp,ϕ ≥b, μΩ
fpLp,ϕ
≥ 1
ϕBx0, N1/nrpB
x0, N1/nr
|x−x0|<N1/nr
b, μΩ
fxpdx
≥ 1
ϕrnpNrn
G\F∩{|x−x0|<N1/nr}
1 2A3Nr
n|x−x 0|−n
p dx
≥ 1
ϕrnpNrn
{A5|F| A2rn1/n<|x−x0|<N1/nr}∩G
1 2A3Nr
n|x−x 0|−n
p dx
ωn−1
ϕrnpNrn
A3Nrn
2
pN1/nr
A5|F| A2rn 1/nt
−pn n−1dt
≥ ωn−1 ϕrnpNr
np−1A3/2p n−np
N1−prn1−p−A51−pn|F| A2rn 1−p
.
2.18
Thus,
|F| A2rn 1−p≤
A6N1−prn1−p
1 ϕrnpfp Lp,ϕ
. 2.19
Now, we claim that
fLp,ϕ≤
C
ϕrn, 2.20
whereCis independent ofr. In fact,
fLp,ϕsup
x∈Rn
t>0
1
ϕ|Bx, t|
1
|Bx, t|
Bx,t
fypdy 1/p
. 2.21
Now, we consider theLp,ϕnorm offin the following two cases.
Case 1t > r. Sinces1/pϕsis nondecreasing ins, then
1
ϕ|Bx, t|
1
|Bx, t|1/p ≤
1
ϕrn
1
Thus,
fLp,ϕ≤sup
x∈Rn
t>0
1
ϕrn
1
rn/p
Bx,t
fypdy 1/p
sup
x∈Rn
t>0
1
ϕrn
1
rn/p
Bx,t∩Bx0,r
fypdy 1/p
≤ C
ϕrn.
2.23
Case 2t≤r. Sinceϕsis nonincreasing ins, then
1
ϕ|Bx, t| ≤
1
ϕrn. 2.24
Thus,
f
Lp,ϕ≤sup
x∈Rn
t>0
1
ϕrn
1
|Bx, t|
Bx,t f
ypdy 1/p
≤ C
ϕrn.
2.25
Now,2.20is established. Then, by2.19and2.20, we get
|F| A2rn≥A7Nrn. 2.26
IfN≤2A−71An2,thenTheorem 1.2is proved. IfN >2A−71An2,then
|F| ≥ A7
2 Nr
n. 2.27
Letgy χBx0,ry. Forx∈F, we have
b, μΩgx≥ |bx| ⎧ ⎪ ⎨ ⎪ ⎩
∞
0
|x−y|≤t
Ωx−y x−yn−1 g
ydy
2
dt t3
⎫ ⎪ ⎬ ⎪ ⎭
1/2
−
⎧ ⎪ ⎨ ⎪ ⎩
∞
0
|x−y|≤t
Ωx−y x−yn−1 b
ygydy 2
dt t3
⎫ ⎪ ⎬ ⎪ ⎭
1/2
:K1−K2.
Noting that ify ∈Bx0, randx∈F, we get|x−y−x−x0| ≤A1. Applying2.11, we
haveΩx−y ≥C1/log2/A1γ. Since|x−y| |x−x0|wheny∈Bx0, randx∈F, it
follows that
K1≥ |bx| ⎧ ⎪ ⎨ ⎪ ⎩
∞
|x−x0| ⎛ ⎜ ⎝
Bx0,r
Ωx−y
x−yn−1 χ{|x−y|≤t}
ydy ⎞ ⎟ ⎠ 2
dt t3
⎫ ⎪ ⎬ ⎪ ⎭
1/2
≥ |bx| ⎛ ⎜ ⎝
∞
|x−x0|
Bx0,r
Ωx−y
x−yn−1 χ{|x−y|≤t}dy dt t3
⎞ ⎟ ⎠
∞
|x−x0|
dt t3
−1/2
≥ C1|bx|
log2/A1
γ|x−x0|
Bx0,r
x−y−n 1
|x−x0|≤t
|x−y|≤t dt t3dy
≥A8|bx||x−x0|−n
Bx0,r
dy
A8rn|bx||x−x0|−n.
2.29
ByΩ∈L∞Sn−1,|x−x
0| |x−y|wheny∈Bx0, randx∈Fand the Minkowski inequality,
we have
K2≤C
Bx0,r b
y x−yndy
≤A9|x−x0|−n
Bx0,r
bydy
A9Nrn|x−x0|−n.
2.30
Thus, by2.28,2.29, and2.30, we get, forx∈F,
b, μΩgx≥A8rn|bx||x−x0|−n−A9Nrn|x−x0|−n. 2.31
ϕNrn≤ϕrn, and|bx|>NA
3/2A4log|x−x0|/rγwhenx∈F, we have
A10
ϕrn ≥gLp,ϕ≥b, μΩ
gLp,ϕ
≥ 1
ϕNrnNrn1/p
|x−x0|<N1/nr
b, μΩ
gxpdx 1/p
≥ 1
ϕrnNrn1/p
|x−x0|<N1/nr
b, μΩgxdx
|x−x0|<N1/nr
dx −1/p
≥ 1
ϕrnNrn
F
b, μΩgxdx
≥ A8rn ϕrnNrn
F
|bx||x−x0|−ndx− A9Nr n
ϕrnNrn
F
|x−x0|−ndx
≥ A11 ϕrn
F
log|x−x0|
r γ
|x−x0|−ndx
− A9 ϕrn
F
|x−x0|−ndx
:L1−L2.
2.32
We first estimateL2.SinceA2r <|x−x0|< N1/nrforx∈F,we have
L2≤
A9ωn−1 ϕrn
N1/nr
A2r
ρ−1dρ≤ A12
ϕrnlogN. 2.33
Now, the estimate ofL1is divided into two cases, namely, 1:γ≥n; 2: 1< γ < n.
Case 1γ ≥ n. Since the function logs/sis decreasing fors≥ 3 and 3r < A2r < |x−x0| < N1/nrforx∈F,by2.27, we get
L1 A11r−n
ϕrn
F log|
x−x0|/r
|x−x0|/r n
log|x−x0|
r
γ−n dx
≥ A7A11
2ϕrn
logA2 γ−nN
logN1/n N1/n
n
≥ A13 ϕrn
logNn.
Case 21 < γ < n. Since the functionlogsγ/sn is decreasing fors ≥ 3 and 3r < A 2r <
|x−x0|< N1/nrforx∈F, by2.27, we have
L1 A11r−n
ϕrn
F
log|x−x0|/r γ
|x−x0|/rn dx
≥ A7A11
2ϕrnN
logN1/nγ N
≥ A14 ϕrn
logNγ.
2.35
From Cases1and2, we know that there exists a constantτ >1 such that
L1≥ A15 ϕrn
logNτ. 2.36
So by2.32,2.33, and2.36, we get
A10≥A15
logNτ−A12logN. 2.37
Then,N≤AΩ, p, n, γ.Theorem 1.2is proved.
3. Proof of
Theorem 1.3
Similar to the proof ofTheorem 1.2, we only give the outline.
Suppose thatb, μSis a bounded operator onLp,ϕRn, we are going to prove that b∈BMORn.
We may assume thatb, μ
SLp,ϕ→Lp,ϕ 1. We want to prove that, for anyx0 ∈Rnand r∈R , the inequality
N 1
|Bx0, r|
Bx0,r b
y−a0dy≤B
Ω, p, n, 3.1
holds, wherea0|Bx0, r|−1
Bx0,rbydy.Sinceb−a0, μ
S b, μS,we may assume that a00.Letfybe as2.3, then2.4–2.8hold. In this proof forj 1, . . . ,13,Bjis a positive
constant depending only onΩ, p,n,, andBi 1 ≤ i < j. SinceΩsatisfies1.2, then there
exists aB1such that 0< B1<1 and
σ
x∈Sn−1: Ωx≥ 2C1
log2/B1 γ
whereσis the measure onSn−1which is induced from the Lebesgue measure onRn. By the
condition1.15, it is easy to see that
Λ:
x∈Sn−1:Ωx≥ 2C1
log2/B1 γ
3.3
is a closed set. As the proof of2.11, we can get the following:
if x∈Λand y∈Sn−1, satisfyingx−y≤B1, thenΩ
y≥ C1
log2/B1
γ. 3.4
TakingB2>3/B1 1, let
Gx∈Rn:|x−x0| ≥B2r, x−x0∈Λ
. 3.5
Forx∈G, we have
b, μ S
fx ⎛ ⎝∞
0
|x−y|<t
|y−z|<t
Ωy−z
y−zn−bx−bzfzdz 2
dydt tn 1 2
⎞ ⎠
1/2
≥
⎛ ⎝∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
×
|y−z|<t
Ωy−z
y−zn−bx−bzfzdz 2
dydt tn 1 2
⎞ ⎠
1/2
≥
⎛ ⎝∞
4|x−x0|
|x−y|<t
2|x−x0|<|y−x0|<3|x−x0|,y−x0∈Λ
×
|y−z|<t
Ωy−z
y−zn−bzfzdz 2
dydt tn 1 2
⎞ ⎠
1/2
− |bx| ⎛ ⎝∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
|y−z|<t
Ωy−z
y−zn−fzdz 2
dydt tn 1 2
⎞ ⎠
1/2
:I1−I2.
ForI1,noting that if|z−x0|< r,|x−x0|> B2|z−x0|, and|y−x0|>2B2|z−x0|,then we get
y−z−y−x0
≤2|z−x0| y−x0 ≤ 1
B2
< B1. 3.7
Then by3.4, we getΩy−z≥C1/log2/B1γ. Since 4|x−x0|>|y−x0| |z−x0| ≥ |y−z| ≥
|y−x0|−|z−x0|>2|x−x0|−|x−x0|/23|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, by2.5,2.7,
2.8, and the H ¨older inequality, we get
I1≥C ∞
4|x−x0|
|x−y|<t,y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
Bx0,r
Ωy−z
y−zn−bzfzχ{|y−z|<t}dz dydt tn 1 2
×
⎛ ⎝∞
4|x−x0|
|x−y|<t,y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
dydt tn 1 2
⎞ ⎠
−1/2
≥C|x−x0|2−n
Bx0,r
bzfz
y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
4|x−x0|<t,|x−y|<t
|y−z|<t
dtdy tn 1 2dz
C|x−x0|2−n
Bx0,r
bzfz
y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
4|x−x0|<t
dtdy tn 1 2dz
≥C|x−x0|−n
Bx0,r
bzfzdz
B3Nrn|x−x0|−n.
3.8
By2.5and2.6, we have
I2|bx| ⎛ ⎝∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
×
Rn
Ωy−z
y−zn−χ{|y−z|<t}−
Ωy−x0
y−x0n−χ{|y−x0|<t}
fzdz 2
dydt tn 1 2
⎞ ⎠
≤ |bx| ⎛ ⎝∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
×
|y−z|<t
|y−x0|<t
Ωy−z y−zn− −
Ωy−x0 y−x0n−
fzdz
2
dydt tn 1 2
⎞ ⎟ ⎠
1/2
|bx| ⎛ ⎜ ⎝
∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
|y−z|<t
|y−x0|≥t
Ωy−z
y−zn−fzdz 2
dydt tn 1 2
⎞ ⎟ ⎠
1/2
|bx| ⎛ ⎜ ⎝
∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
|y−z|≥t
|y−x0|<t
Ωy−x0 y−x0n−
fzdz 2
dydt tn 1 2
⎞ ⎟ ⎠
1/2
:I21 I22 I23.
3.9
InI22,we havet ≤ |y−x0| < 3|x−x0|andt ≥ 4|x−x0|. InI23,we gett ≤ |y−z| < 4|x−x0|
andt≥4|x−x0|.It is easy to see thatI22 I230.Now, we estimateI21,byΩ∈L∞Sn−1, the
Minkowski inequality,Lemma 2.1for|y−x0|>2|z−x0|, and2.4, we get
I21 ≤C|bx|
Bx0,r
fzdz ⎛ ⎝
2|x−x0|<|y−x0|<3|x−x0|
4|x−x0|≤t,|y−z|<t
|y−x0|<t,|x−y|≤t
× 1
y−x02n−
logy−x0/r 2γ
dtdy tn 1 2
1/2
≤B4|bx|rn|x−x0|−n
log|x−x0|
r −γ
.
3.10
From3.9and3.10, we get
I2≤B4|bx|rn|x−x0|−n
log|x−x0|
r −γ
. 3.11
Let
F !
x∈G:|bx|> B3N
2B4
log|x−x0|
r γ
, |x−x0|< N1/nr "
Without loss of generality, we may assume thatN > B2 > 1, otherwise, we get the desired
result. Sinceϕtis nonincreasing, we haveϕ|Bx0, N1/nr|≤ϕ|Bx0, r| ϕrn. Then by,
3.6,3.8, and3.11, we get
fpLp,ϕ≥b, μS
fpLp,ϕ
≥ 1
ϕBx0, N1/nrpB
x0, N1/nr
|x−x0|<N1/nr
b, μSfxpdx
≥ 1
ϕrnpNrn
G\F∩{|x−x0|<N1/nr}
1 2B3Nr
n|x−x 0|−n
p dx
≥ 1
ϕrnpNrn
{B5|F| B2rn1/n<|x−x0|<N1/nr}∩G
1 2B3Nr
n|x−x 0|−n
p dx
1
ϕrnpNrn
B3Nrn
2
pN1/nr
B5|F| B2rn1/n
t−pn n−1dt
ΛJ
xdσx
≥ 1
ϕrnpσΛNr
np−1B3/2p n−np
N1−prn1−p−B51−pn|F| B2rn 1−p
.
3.13
Thus,
|F| B2rn 1−p≤
B6N1−prn1−p
1 ϕrnpfpLp,λ
. 3.14
Then, by2.20and3.14, we get
|F| B2rn≥B7Nrn. 3.15
IfN≤2B−1
7 B2n,thenTheorem 1.3is proved. IfN >2B−71B2n,then
|F| ≥ B7
2 Nr
Letgy χBx0,ry. Forx∈F, we have
b, μSgx ⎛ ⎝∞
0
|x−y|<t
|y−z|<t
Ωy−z
y−zn−bx−bzgzdz 2
dydt tn 1 2
⎞ ⎠
1/2
≥
⎛ ⎝∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
|y−z|<t
Ωy−z
y−zn−bx−bzgzdz 2
dydt tn 1 2
⎞ ⎠
1/2
≥|bx| ⎛ ⎝∞
4|x−x0|
|x−y|<t
2|x−x0|<|y−x0|<3|x−x0|,y−x0∈Λ
|y−z|<t
Ωy−z y−zn−gzdz
2
dydt tn 1 2
⎞ ⎠
1/2
−
⎛ ⎝∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
|y−z|<t
Ωy−z
y−zn−bzgzdz 2
dydt tn 1 2
⎞ ⎠
1/2
:K1−K2.
3.17
ForK1,as above mentioned, we haveΩy−z≥C1/log2/B1γ.Since 4|x−x0| ≥ |y−z| ≥
3|x−x0|/2 and 4|x−x0|>|x−y|>|x−x0|, it follows the H ¨older inequality that
K1|bx| ⎧ ⎨ ⎩
∞
4|x−x0|
|x−y|<t,y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
Bx0,r
Ωy−z
y−zn−χ{|y−z|<t}dz 2
dydt tn 1 2
⎫ ⎬ ⎭
1/2
≥ |bx| ∞
4|x−x0|
|x−y|<t,y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
Bx0,r
Ωy−z
y−zn−χ{|y−z|<t}dz dydt tn 1 2
×
⎛ ⎝∞
4|x−x0|
|x−y|<t,y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
dydt tn 1 2
⎞ ⎠
−1/2
≥C|bx||x−x0|2−n
Bx0,r
y−x0∈Λ 2|x−x0|<|y−x0|<3|x−x0|
4|x−x0|<t
dtdy tn 1 2dz
≥B8N|x−x0|−n
Bx0,r
dz
B8Nrn|x−x0|−n.
ByΩ∈L∞Sn−1, the Minkowski inequality, and|x−x
0| |y−z|for 2|x−x0|<|y−x0|<3|x−x0|
andz∈Bx0, r, we get
K2 ⎛ ⎝∞
4|x−x0|
|x−y|<t 2|x−x0|<|y−x0|<3|x−x0|
Bx0,r
Ωy−z
y−zn−bzχ{|y−z|<t}dz 2
dydt tn 1 2
⎞ ⎠
1/2
≤ C
|x−x0|n−
Bx0,r
|bz|dz
2|x−x0|<|y−x0|<3|x−x0| ∞
4|x−x0|
dtdy tn 1 2
1/2
≤B9|x−x0|−n
Bx0,r
|bz|dz
B9Nrn|x−x0|−n.
3.19
Thus, by3.17,3.18, and3.19, we get, forx∈F,
b, μ S
gx≥B8rn|bx||x−x0|−n−B9Nrn|x−x0|−n. 3.20
Thus, by3.20,ϕNrn ≤ ϕrn,|bx| >NB
3/2B4log|x−x0|/rγ whenx ∈F and the
H ¨older inequality, we have
B10
ϕrn ≥gLp,ϕ ≥b, μS
gLp,ϕ
≥ 1
ϕNrnNrn1/p
|x−x0|<N1/nr b, μ
S
gxpdx 1/p
≥ 1
ϕrnNrn1/p
|x−x0|<N1/nr b, μ
S
gxdx
|x−x0|<N1/nr
dx −1/p
≥ 1
ϕrnNrn
F b, μ
S
gxdx
≥ B8rn ϕrnNrn
F
|bx||x−x0|−ndx−
B9Nrn ϕrnNrn
F
|x−x0|−ndx
≥ B11 ϕrn
F
log|x−x0|
r γ
|x−x0|−ndx
− B9 ϕrn
F
|x−x0|−ndx
:L1−L2.
As the proof of2.33and2.36, we can get that there exists a constantτ >1 such that
L1≥ B12 ϕrn
logNτ,
L2≤ B13
ϕrnlogN.
3.22
So, by3.21and3.22, we get
B10 ≥B12
logNτ−B13logN. 3.23
Then,N≤BΩ, p, n, .Theorem 1.3is 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 ChinaGrant nos.: 10901017, 10931001, SRFDP of ChinaGrant no.: 20090003110018, and NSF of ZhenjiangGrant no.: Y7080325.
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