Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013, Article ID 257537, 7 pageshttp://dx.doi.org/10.1155/2013/257537
Research ArticleHigher Order Commutators of Fractional Integral Operator onthe Homogeneous Herz Spaces with Variable Exponent
Liwei Wang,1 Meng Qu,2 and Lisheng Shu2
1 School of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, China2 School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China
Correspondence should be addressed to Liwei Wang; [email protected]
Received 28 March 2013; Accepted 20 May 2013
Academic Editor: Dachun Yang
Copyright © 2013 Liwei Wang 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.
By decomposing functions, we establish estimates for higher order commutators generated by fractional integral with BMOfunctions or the Lipschitz functions on the homogeneous Herz spaces with variable exponent.These estimates extend some knownresults in the literatures.
1. Introduction
Let 𝑏 be a locally integrable function, 0 < 𝛽 < 𝑛, and 𝑚 ∈ N;the higher order commutators of fractional integral operator𝐼𝑚
𝛽,𝑏are defined by
𝐼𝑚
𝛽,𝑏𝑓 (𝑥) = ∫
R𝑛
[𝑏 (𝑥) − 𝑏 (𝑦)]𝑚
𝑥 − 𝑦𝑛−𝛽
𝑓 (𝑦) 𝑑𝑦. (1)
Obviously, 𝐼0𝛽,𝑏
= 𝐼𝛽and 𝐼1
𝛽,𝑏= [𝑏, 𝐼
𝛽]. The famous
Hardy-Littlewood-Sobolev theorem tells us that the frac-tional integral operator 𝐼
𝛽is a bounded operator from the
usual Lebesgue spaces 𝐿𝑝1(R𝑛) to 𝐿𝑝2(R𝑛) when 0 < 𝑝1
<
𝑝2
< ∞ and 1/𝑝1− 1/𝑝
2= 𝛽/𝑛. Also, many generalized
results about 𝐼𝛽and the commutator [𝑏, 𝐼
𝛽] on some function
spaces have been studied; see [1–3] for details.It is well known that the main motivation for studying
the spaces with variable exponent arrived in the nonlinearelasticity theory and differential equations with nonstandardgrowth. Since the fundamental paper [4] by Kováčik andRákosnı́k appeared in 1991, the Lebesgue spaces with variableexponent 𝐿𝑝(⋅)(R𝑛) have been extensively investigated. In therecent twenty years, boundedness of some important operat-ors, for example, the Calderón-Zygmund operators, frac-tional integrals, and commutators, on 𝐿𝑝(⋅)(R𝑛) has beenobtained; see [5–7]. Recently, Diening [8] extended the
(𝐿𝑝1(R𝑛), 𝐿𝑝2(R𝑛)) boundedness of 𝐼
𝛽to the Lebesgue spaces
with variable exponent. Izuki [7] first introduced the Herzspaces with variable exponent �̇�𝛼,𝑞
𝑝(⋅)(R𝑛), which is a general-
ized space of the Herz space �̇�𝛼,𝑞𝑝
(R𝑛); see [9, 10], and in caseof 𝑏 ∈ BMO(R𝑛), he obtained the boundedness propertiesof the commutator [𝑏, 𝐼
𝛽]. The paper [11] by Lu et al. indi-
cates that the commutator [𝑏, 𝐼𝛽] with 𝑏 ∈ BMO(R𝑛) and
with 𝑏 ∈ Lip𝛼(R𝑛) (0 < 𝛼 ≤ 1) has many different pro-
perties. In 2012, Zhou [12] studied the boundedness of 𝐼𝛽
on the Herz spaces with variable exponent and proved thatthe boundedness properties of the commutator [𝑏, 𝐼
𝛽] also
hold in case of 𝑏 ∈ Lip𝛼(R𝑛) (0 < 𝛼 ≤ 1). The higher
order commutators 𝐼𝑚𝛽,𝑏
are recently considered byWang et al.in the paper [13, 14]; they established the BMO and theLipschitz estimates for 𝐼𝑚
𝛽,𝑏on the Lebesgue spaces with vari-
able exponent 𝐿𝑝(⋅)(R𝑛). Motivated by [7, 12–14], in thisnote, we establish the boundedness of the higher order com-mutators 𝐼𝑚
𝛽,𝑏on the Herz spaces with variable exponent.
For brevity, |𝐸| denotes the Lebesgue measure for ameasurable set 𝐸 ⊂ R𝑛, and 𝑓
𝐸denotes the mean value of 𝑓
on 𝐸 (𝑓𝐸= (1/|𝐸|) ∫
𝐸
𝑓(𝑥)𝑑𝑥). The exponent 𝑝(⋅)means theconjugate of 𝑝(⋅), that is, 1/𝑝(⋅)+1/𝑝(⋅) = 1.𝐶 denotes a pos-itive constant, which may have different values even in thesame line. Let us first recall some definitions and nota-tions.
CORE Metadata, citation and similar papers at core.ac.uk
Provided by MUCC (Crossref)
https://core.ac.uk/display/186894263?utm_source=pdf&utm_medium=banner&utm_campaign=pdf-decoration-v1
2 Journal of Function Spaces and Applications
Definition 1. For 0 < 𝛾 ≤ 1, the Lipschitz space Lip𝛾(R𝑛) is
the space of functions 𝑓 satisfying
𝑓Lip𝛾
= sup𝑥,𝑦∈R𝑛,𝑥 ̸= 𝑦
𝑓 (𝑥) − 𝑓 (𝑦)
𝑥 − 𝑦𝛾
< ∞. (2)
Definition 2. For𝑓 ∈ 𝐿1loc(R𝑛
), the boundedmean oscillationspace BMO(R𝑛) is the space of functions 𝑓 satisfying
𝑓BMO = sup
𝐵
1
|𝐵|∫𝐵
𝑓 (𝑥) − 𝑓𝐵 𝑑𝑥 < ∞, (3)
where the supremum is taken over all balls 𝐵 in R𝑛.
Definition 3. Let 𝑝(⋅) : 𝐸 → [1,∞) be a measurable func-tion.
(1) The Lebesgue space with variable exponent 𝐿𝑝(⋅)(𝐸) isdefined by
𝐿𝑝(⋅)
(𝐸) = {𝑓 is measurable : ∫𝐸
(
𝑓 (𝑥)
𝜆)
𝑝(𝑥)
𝑑𝑥
< ∞ for some constant 𝜆 > 0} .
(4)
(2) The space with variable exponent 𝐿𝑝(⋅)loc (𝐸) is definedby
𝐿𝑝(⋅)
loc (𝐸)
= {𝑓 : 𝑓 ∈ 𝐿𝑝(⋅)
(𝐾) for all compact subsets 𝐾 ⊂ 𝐸 } .(5)
The Lebesgue space 𝐿𝑝(⋅)(𝐸) is a Banach space with theLuxemburg norm
𝑓𝐿𝑝(⋅)(𝐸) = inf {𝜆 > 0 : ∫
𝐸
(
𝑓 (𝑥)
𝜆)
𝑝(𝑥)
𝑑𝑥 ≤ 1} . (6)
We denote
𝑝−= ess inf {𝑝 (𝑥) : 𝑥 ∈ 𝐸} ,
𝑝+= ess sup {𝑝 (𝑥) : 𝑥 ∈ 𝐸} ,
P (𝐸) = {𝑝 (⋅) : 𝑝−> 1, 𝑝
+< ∞} ,
B (𝐸) = {𝑝 (⋅) : 𝑝 (⋅) ∈ P (𝐸) ,
𝑀 is bounded on 𝐿𝑝(⋅) (𝐸)} ,
(7)
where the Hardy-Littlewood maximal operator 𝑀 is definedby
𝑀𝑓(𝑥) = sup𝑟>0
𝑟−𝑛
∫𝐵(𝑥,𝑟)∩𝐸
𝑓 (𝑦) 𝑑𝑦, (8)
where 𝐵(𝑥, 𝑟) = {𝑦 ∈ R𝑛 : |𝑥 − 𝑦| < 𝑟}.
Proposition 4 (see [15]). If 𝑝(⋅) ∈ P(𝐸) satisfies
𝑝 (𝑥) − 𝑝 (𝑦) ≤
−𝐶
log (𝑥 − 𝑦),
𝑥 − 𝑦 ≤
1
2,
𝑝 (𝑥) − 𝑝 (𝑦) ≤
𝐶
log (𝑒 + |𝑥|),
𝑦 ≤ |𝑥| ,
(9)
then one has 𝑝(⋅) ∈ B(𝐸).
Let 𝐵𝑘= {𝑥 ∈ R𝑛 : |𝑥| ⩽ 2𝑘}, 𝑅
𝑘= 𝐵𝑘\𝐵𝑘−1
, and 𝜒𝑘= 𝜒𝑅𝑘
be the characteristic function of the set 𝑅𝑘for 𝑘 ∈ Z. For
𝑚 ∈ N, we denote 𝜒𝑚
= 𝜒𝑅𝑚
if𝑚 ≥ 1, and 𝜒0= 𝜒𝐵0
.
Definition 5 (see [7]). For 𝛼 ∈ R, 0 < 𝑞 ≤ ∞ and 𝑝(⋅) ∈P(R𝑛).
(1) The homogeneous Herz spaces �̇�𝛼,𝑞𝑝(⋅)
(R𝑛) are definedby
�̇�𝛼,𝑞
𝑝(⋅)(R𝑛
) = {𝑓 ∈ 𝐿𝑝(⋅)
loc (R𝑛
\ {0}) :𝑓
�̇�𝛼,𝑞
𝑝(⋅)(R𝑛)
< ∞} , (10)
where𝑓
�̇�𝛼,𝑞
𝑝(⋅)(R𝑛)
={2𝛼𝑘𝑓𝜒𝑘
𝐿𝑝(⋅)(R𝑛)}∞
𝑘=−∞
ℓ𝑞(Z). (11)
(2) The nonhomogeneous Herz spaces 𝐾𝛼,𝑞𝑝(⋅)
(R𝑛) are de-fined by
𝐾𝛼,𝑞
𝑝(⋅)(R𝑛
) = {𝑓 ∈ 𝐿𝑝(⋅)
loc (R𝑛
) :𝑓
𝐾𝛼,𝑞
𝑝(⋅)(R𝑛)
< ∞} , (12)
where𝑓
𝐾𝛼,𝑞
𝑝(⋅)(R𝑛)
={2𝛼𝑚𝑓𝜒𝑚
𝐿𝑝(⋅)(R𝑛)}∞
𝑚=0
ℓ𝑞(N). (13)
In this note, we obtain the following results.
Theorem 6. Suppose that 𝑏 ∈ Lip𝛽1
(R𝑛) (0 < 𝛽1
< 1),𝑝2(⋅) ∈ P(R𝑛) satisfies conditions (9) in Proposition 4. If
0 < 𝑟 < min {1/(𝑝1)+, 1/(𝑝
2)+}, 0 < 𝛽 + 𝑚𝛽
1< 𝑛𝑟, 0 <
𝛼 < 𝑛𝑟 − 𝛽 − 𝑚𝛽1, 0 < 𝑞
1≤ 𝑞2
< ∞, and 1/𝑝1(𝑥) −
1/𝑝2(𝑥) = (𝛽 + 𝑚𝛽
1)/𝑛, then the higher order commutators
𝐼𝑚
𝛽,𝑏are bounded from �̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛) to �̇�𝛼,𝑞2
𝑝2(⋅)(R𝑛).
Theorem 7. Suppose that 𝑏 ∈ BMO(Rn), 𝑝2(⋅) ∈ P(R𝑛)
satisfies conditions (9) in Proposition 4. If 0 < 𝑟 <min {1/(𝑝
1)+, 1/(𝑝
2)+}, 0 < 𝛽 < 𝑛𝑟, 0 < 𝛼 < 𝑛𝑟 − 𝛽,
0 < 𝑞1
≤ 𝑞2
< ∞, and 1/𝑝1(𝑥) − 1/𝑝
2(𝑥) = 𝛽/𝑛, then the
higher order commutators 𝐼𝑚𝛽,𝑏
are bounded from �̇�𝛼,𝑞1𝑝1(⋅)(R𝑛) to
�̇�𝛼,𝑞2
𝑝2(⋅)(R𝑛).
Remark A. The previous main results generalize the(𝐿𝑝(⋅)
(R𝑛), 𝐿𝑞(⋅)(R𝑛)) boundedness of the higher ordercommutators 𝐼𝑚
𝛽,𝑏in [13] to the case of the Herz spaces with
variable exponent. If 𝑚 = 1, our conclusions coincide withthe corresponding results in [7, 12]. Moreover, the sameboundedness also holds for the nonhomogeneous case.
Journal of Function Spaces and Applications 3
2. Proof of Theorems 6 and 7
To prove our main results, we need the following lemmas.
Lemma 8 (see [4]). Let 𝑝(⋅) ∈ P(R𝑛); if 𝑓 ∈ 𝐿𝑝(⋅)(R𝑛) and𝑔 ∈ 𝐿𝑝
(⋅)
(R𝑛), then
∫R𝑛
𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥 ≤ 𝑟𝑝
𝑓𝐿𝑝(⋅)(R𝑛)
𝑔𝐿𝑝(⋅)(R𝑛)
, (14)
where 𝑟𝑝= 1 + 1/𝑝
−− 1/𝑝
+.
Lemma 9 (see [7]). Let 𝑝(⋅) ∈ B(R𝑛); then for all balls 𝐵 inR𝑛,
1
|𝐵|
𝜒𝐵𝐿𝑝(⋅)(R𝑛)
𝜒𝐵𝐿𝑝(⋅)(R𝑛)
≤ 𝐶. (15)
Lemma 10 (see [7]). Let 𝑝2(⋅) ∈ B(R𝑛); then for all balls 𝐵 in
R𝑛 and all measurable subsets 𝑆 ⊂ 𝐵, one can take a constant0 < 𝑟 < 1/(𝑝
2)+, so that
𝜒𝑆𝐿𝑝
2(⋅)
(R𝑛)𝜒𝐵
𝐿𝑝
2(⋅)
(R𝑛)
≤ 𝐶(|𝑆|
|𝐵|)
𝑟
. (16)
Lemma 11 (see [8]). Suppose that 𝑝1(⋅) ∈ P(R𝑛) satisfies
conditions (9) in Proposition 4, 0 < 𝛽 < 𝑛/(𝑝1)+and 1/𝑝
1(𝑥)−
1/𝑝2(𝑥) = 𝛽/𝑛; then
𝐼𝛽(𝑓)
𝐿𝑝2(⋅)(R𝑛)≤ 𝐶
𝑓𝐿𝑝1(⋅)(R𝑛). (17)
Lemma 12 (see [13]). Suppose that 𝑝1(⋅), 𝑝2(⋅) ∈ P(R𝑛).
(1) Let 0 < 𝛽 < 𝑛/(𝑝1)+, 𝑏 ∈ BMO(Rn). If 𝑝
2(⋅) satisfies
conditions (9) in Proposition 4 and 1/𝑝1(𝑥)−1/𝑝
2(𝑥) =
𝛽/𝑛, then𝐼𝑚
𝛽,𝑏(𝑓)
𝐿𝑝2(⋅)(R𝑛)≤ 𝐶‖𝑏‖
𝑚
BMO𝑓
𝐿𝑝1(⋅)(R𝑛). (18)
(2) Let 0 < 𝛽 + 𝑚𝛽1< 𝑛/(𝑝
1)+, 𝑏 ∈ Lip
𝛽1
(R𝑛) (0 < 𝛽1<
1). If 𝑝2(⋅) satisfies conditions (9) in Proposition 4 and
1/𝑝1(𝑥) − 1/𝑝
2(𝑥) = (𝛽 + 𝑚𝛽
1)/𝑛, then
𝐼𝑚
𝛽,𝑏(𝑓)
𝐿𝑝2(⋅)(R𝑛)≤ 𝐶‖𝑏‖
𝑚
Lip𝛽1
𝑓𝐿𝑝1(⋅)(R𝑛). (19)
Lemma 13 (see [16]). Let 𝑏 ∈ BMO(Rn), 𝑘 > 𝑗 (𝑘, 𝑗 ∈ N);one has
(1) 𝐶−1||𝑏||𝑚BMO ≤ sup𝐵⊂R𝑛(1/||𝜒𝐵||𝐿𝑝(⋅)(R𝑛))||(𝑏 −𝑏𝐵)𝑚
𝜒𝐵||𝐿𝑝(⋅)(R𝑛) ≤ 𝐶||𝑏||
𝑚
BMO;(2) ||(𝑏 − 𝑏
𝐵𝑗
)𝑚
𝜒𝐵𝑘
||𝐿𝑝(⋅)(R𝑛) ≤ 𝐶(𝑘 − 𝑗)
𝑚
||𝑏||𝑚
BMO×||𝜒𝐵𝑘
||𝐿𝑝(⋅)(R𝑛).
Proof of Theorem 6. Let 𝑓 ∈ �̇�𝛼,𝑞1𝑝1(⋅)(R𝑛); we can write
𝑓 (𝑥) =
∞
∑
𝑗=−∞
𝑓 (𝑥) 𝜒𝑗(𝑥) =
∞
∑
𝑗=−∞
𝑓𝑗(𝑥) . (20)
For 0 < 𝑞1/𝑞2≤ 1, applying the inequality
(
∞
∑
𝑖=1
𝑎𝑖)
𝑞1/𝑞2
≤
∞
∑
𝑖=1
𝑎𝑞1/𝑞2
𝑖(𝑎𝑖> 0, 𝑖 = 1, 2 . . .) , (21)
we obtain𝐼𝑚
𝛽,𝑏(𝑓)
𝑞1
�̇�𝛼,𝑞2
𝑝2(⋅)(R𝑛)
= 𝐶(
∞
∑
𝑘=−∞
2𝛼𝑞2𝑘𝐼𝑚
𝛽,𝑏(𝑓) 𝜒𝑘
𝑞2
𝐿𝑝2(⋅)(R𝑛)
)
𝑞1/𝑞2
≤ 𝐶
∞
∑
𝑘=−∞
2𝛼𝑞1𝑘
(
𝑘−2
∑
𝑗=−∞
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛))
𝑞1
+ 𝐶
∞
∑
𝑘=−∞
2𝛼𝑞1𝑘
(
∞
∑
𝑗=𝑘−1
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛))
𝑞1
= 𝑈1+ 𝑈2.
(22)
We first estimate 𝑈1. Noting that if 𝑥 ∈ 𝑅
𝑘, 𝑦 ∈ 𝑅
𝑗, and
𝑗 ≤ 𝑘 − 2, then |𝑥 − 𝑦| ∼ |𝑥| ∼ 2𝑘, we get
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
≤ ∫𝑅𝑗
𝑏 (𝑥) − 𝑏 (𝑦)𝑚
𝑥 − 𝑦𝑛−𝛽
𝑓𝑗(𝑦)
𝑑𝑦 ⋅ 𝜒
𝑘(𝑥)
≤ 𝐶2𝑘(𝛽−𝑛)
∫𝑅𝑗
𝑏 (𝑥)− 𝑏 (𝑦)𝑚
𝑓𝑗(𝑦)
𝑑𝑦 ⋅ 𝜒
𝑘(𝑥)
≤ 𝐶2𝑘(𝛽+𝑚𝛽
1−𝑛)
‖𝑏‖𝑚
Lip𝛽1
∫𝑅𝑗
𝑓𝑗(𝑦)
𝑑𝑦 ⋅ 𝜒
𝑘(𝑥) .
(23)
By Hölder’s inequality, Lemmas 9 and 10, we have𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛)
≤ 𝐶2𝑘(𝛽+𝑚𝛽
1−𝑛)
‖𝑏‖𝑚
Lip𝛽1
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×𝜒𝐵𝑘
𝐿𝑝2(⋅)(R𝑛)
𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
≤ 𝐶2𝑘(𝛽+𝑚𝛽
1)
‖𝑏‖𝑚
Lip𝛽1
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
𝜒𝐵𝑘
−1
𝐿𝑝
2(⋅)
(R𝑛)
≤ 𝐶2𝑘(𝛽+𝑚𝛽
1)
‖𝑏‖𝑚
Lip𝛽1
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
𝜒𝐵𝑗
−1
𝐿𝑝
2(⋅)
(R𝑛)
𝜒𝐵𝑗
𝐿𝑝
2(⋅)
(R𝑛)𝜒𝐵𝑘
𝐿𝑝
2(⋅)
(R𝑛)
≤ 𝐶2𝑘(𝛽+𝑚𝛽
1)
2𝑛𝑟(𝑗−𝑘)
‖𝑏‖𝑚
Lip𝛽1
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
𝜒𝐵𝑗
−1
𝐿𝑝
2(⋅)
(R𝑛).
(24)
4 Journal of Function Spaces and Applications
Note that
𝐼𝛽+𝑚𝛽
1
(𝜒𝐵𝑗
) (𝑥) ≥ 𝐼𝛽+𝑚𝛽
1
(𝜒𝐵𝑗
) (𝑥) ⋅ 𝜒𝐵𝑗
(𝑥)
= ∫𝐵𝑗
𝑑𝑦
𝑥 − 𝑦𝑛−𝛽−𝑚𝛽
1
⋅ 𝜒𝐵𝑗
(𝑥)
≥ 𝐶2𝑗(𝛽+𝑚𝛽
1)
⋅ 𝜒𝐵𝑗
(𝑥) .
(25)
By Lemmas 8 and 11, we obtain
𝜒𝐵𝑗
−1
𝐿𝑝
2(⋅)
(R𝑛)≤ 𝐶2−𝑛𝑗
𝜒𝐵𝑗
𝐿𝑝2(⋅)(R𝑛)
≤ 𝐶2−𝑛𝑗
2−𝑗(𝛽+𝑚𝛽
1)𝐼𝛽+𝑚𝛽
1
(𝜒𝐵𝑗
)𝐿𝑝2(⋅)(R𝑛)
≤ 𝐶2−𝑗(𝛽+𝑚𝛽
1)
2−𝑛𝑗
𝜒𝐵𝑗
𝐿𝑝1(⋅)(R𝑛)
≤ 𝐶2−𝑗(𝛽+𝑚𝛽
1)𝜒𝐵𝑗
−1
𝐿𝑝
1(⋅)
(R𝑛).
(26)
Combining (24) and (26), we have the estimate
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛)≤ 𝐶2(𝑘−𝑗)(𝛽+𝑚𝛽
1−𝑛𝑟)
× ‖𝑏‖𝑚
Lip𝛽1
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛).
(27)
Thus,
𝑈1≤ 𝐶‖𝑏‖
𝑚𝑞1
Lip𝛽1
×
∞
∑
𝑘=−∞
(
𝑘−2
∑
𝑗=−∞
2𝛼𝑗𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)2(𝑘−𝑗)(𝛽+𝑚𝛽
1−𝑛𝑟+𝛼)
)
𝑞1
.
(28)
If 1 < 𝑞1
< ∞, noting that 𝛽 + 𝑚𝛽1− 𝑛𝑟 + 𝛼 < 0, by
Hölder’s inequality, we have
𝑈1≤ 𝐶‖𝑏‖
𝑚𝑞1
Lip𝛽1
×
∞
∑
𝑘=−∞
(
𝑘−2
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
2(𝑘−𝑗)(𝛽+𝑚𝛽
1−𝑛𝑟+𝛼)𝑞
1/2
)
× (
𝑘−2
∑
𝑗=−∞
2(𝑘−𝑗)(𝛽+𝑚𝛽
1−𝑛𝑟+𝛼)𝑞
1/2
)
𝑞1/𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
∞
∑
𝑘=𝑗+2
2(𝑘−𝑗)(𝛽+𝑚𝛽
1−𝑛𝑟+𝛼)𝑞
1/2
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
𝑓𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(29)
If 0 < 𝑞1≤ 1, by inequality (21), we have
𝑈1≤ 𝐶‖𝑏‖
𝑚𝑞1
Lip𝛽1
×
∞
∑
𝑘=−∞
𝑘−2
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
2(𝑘−𝑗)(𝛽+𝑚𝛽
1−𝑛𝑟+𝛼)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
∞
∑
𝑘=𝑗+2
2(𝑘−𝑗)(𝛽+𝑚𝛽
1−𝑛𝑟+𝛼)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
𝑓𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(30)
Next, we estimate 𝑈2. By Lemma 12(2), we obtain
𝑈2≤ 𝐶‖𝑏‖
𝑚𝑞1
Lip𝛽1
∞
∑
𝑘=−∞
(
∞
∑
𝑗=𝑘−1
2𝛼𝑗𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)2𝛼(𝑘−𝑗)
)
𝑞1
. (31)
If 1 < 𝑞1< ∞, by Hölder’s inequality, we have
𝑈2≤ 𝐶‖𝑏‖
𝑚𝑞1
Lip𝛽1
∞
∑
𝑘=−∞
(
∞
∑
𝑗=𝑘−1
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
2𝛼(𝑘−𝑗)𝑞
1/2
)
× (
∞
∑
𝑗=𝑘−1
2𝛼(𝑘−𝑗)𝑞
1/2
)
𝑞1/𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
𝑗+1
∑
𝑘=−∞
2𝛼(𝑘−𝑗)𝑞
1/2
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
𝑓𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(32)
If 0 < 𝑞1≤ 1, by inequality (21), we have
𝑈2≤ 𝐶‖𝑏‖
𝑚𝑞1
Lip𝛽1
∞
∑
𝑘=−∞
∞
∑
𝑗=𝑘−1
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
2𝛼(𝑘−𝑗)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
𝑗+1
∑
𝑘=−∞
2𝛼(𝑘−𝑗)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
Lip𝛽1
𝑓𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(33)
Combining the estimates for 𝑈1and 𝑈
2, the proof of
Theorem 6 is completed.
Proof of Theorem 7. Let 𝑓 ∈ �̇�𝛼,𝑞1𝑝1(⋅)(R𝑛); we can write
𝑓 (𝑥) =
∞
∑
𝑗=−∞
𝑓 (𝑥) 𝜒𝑗(𝑥) =
∞
∑
𝑗=−∞
𝑓𝑗(𝑥) . (34)
Journal of Function Spaces and Applications 5
By inequality (21), we obtain
𝐼𝑚
𝛽,𝑏(𝑓)
𝑞1
�̇�𝛼,𝑞2
𝑝2(⋅)(R𝑛)
= 𝐶(
∞
∑
𝑘=−∞
2𝛼𝑞2𝑘𝐼𝑚
𝛽,𝑏(𝑓) 𝜒𝑘
𝑞2
𝐿𝑝2(⋅)(R𝑛)
)
𝑞1/𝑞2
≤ 𝐶
∞
∑
𝑘=−∞
2𝛼𝑞1𝑘
(
𝑘−2
∑
𝑗=−∞
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛))
𝑞1
+ 𝐶
∞
∑
𝑘=−∞
2𝛼𝑞1𝑘
(
∞
∑
𝑗=𝑘−1
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛))
𝑞1
= 𝑉1+ 𝑉2.
(35)
For 𝑉1, using Hölder’s inequality and Lemma 8, we have
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
≤ 𝐶2𝑘(𝛽−𝑛)
∫𝑅𝑗
𝑏 (𝑥) − 𝑏 (𝑦)𝑚
𝑓𝑗(𝑦)
𝑑𝑦 ⋅ 𝜒
𝑘(𝑥)
≤ 𝐶2𝑘(𝛽−𝑛)
𝑚
∑
𝑖=0
𝐶𝑖
𝑚
𝑏 (𝑥) − 𝑏
𝐵𝑗
𝑚−𝑖
× ∫𝑅𝑗
𝑏𝐵𝑗
− 𝑏 (𝑦)
𝑖 𝑓𝑗(𝑦)
𝑑𝑦
≤ 𝐶2𝑘(𝛽−𝑛)
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×
𝑚
∑
𝑖=0
𝐶𝑖
𝑚
𝑏 (𝑥) − 𝑏
𝐵𝑗
𝑚−𝑖(𝑏𝐵𝑗
− 𝑏)𝑖
𝜒𝑗
𝐿𝑝
1(⋅)
(R𝑛).
(36)
By Lemmas 9, 10, and 13, we have
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛)
≤ 𝐶2𝑘(𝛽−𝑛)
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×
𝑚
∑
𝑖=0
𝐶𝑖
𝑚
(𝑏 (𝑥) − 𝑏
𝐵𝑗
)𝑚−𝑖
𝜒𝑘
𝐿𝑝2(⋅)(R𝑛)
×
(𝑏𝐵𝑗
− 𝑏)𝑖
𝜒𝑗
𝐿𝑝
1(⋅)
(R𝑛)
≤ 𝐶2𝑘(𝛽−𝑛)
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×
𝑚
∑
𝑖=0
𝐶𝑖
𝑚(𝑘 − 𝑗)
𝑚−𝑖
‖𝑏‖𝑚−𝑖
BMO𝜒𝐵𝑘
𝐿𝑝2(⋅)(R𝑛)
× ‖𝑏‖𝑖
BMO𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
= 𝐶(𝑘 − 𝑗 + 1)𝑚
‖𝑏‖𝑚
BMO𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
× 2𝑘(𝛽−𝑛)
𝜒𝐵𝑘
𝐿𝑝2(⋅)(R𝑛)
𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
≤ 𝐶(𝑘 − 𝑗 + 1)𝑚
‖𝑏‖𝑚
BMO𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
× 2𝑘𝛽𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
𝜒𝐵𝑘
−1
𝐿𝑝
2(⋅)
(R𝑛)
≤ 𝐶(𝑘 − 𝑗 + 1)𝑚
‖𝑏‖𝑚
BMO𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
× 2𝑘𝛽𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
𝜒𝐵𝑘
−1
𝐿𝑝
2(⋅)
(R𝑛)
𝜒𝐵𝑗
𝐿𝑝
2(⋅)
(R𝑛)𝜒𝐵𝑘
𝐿𝑝
2(⋅)
(R𝑛)
≤ 𝐶(𝑘 − 𝑗 + 1)𝑚
‖𝑏‖𝑚
BMO𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
× 2𝑘𝛽
2𝑛𝑟(𝑗−𝑘)
𝜒𝐵𝑗
𝐿𝑝
1(⋅)
(R𝑛)
𝜒𝐵𝑗
−1
𝐿𝑝
2(⋅)
(R𝑛).
(37)
Note that
𝐼𝛽(𝜒𝐵𝑗
) (𝑥) ≥ ∫𝐵𝑗
𝑑𝑦
𝑥 − 𝑦𝑛−𝛽
⋅ 𝜒𝐵𝑗
(𝑥) ≥ 𝐶2𝑗𝛽
⋅ 𝜒𝐵𝑗
(𝑥) .
(38)
By Lemmas 8 and 11, we obtain
𝜒𝐵𝑗
−1
𝐿𝑝
2(⋅)
(R𝑛)≤ 𝐶2−𝑛𝑗
𝜒𝐵𝑗
𝐿𝑝2(⋅)(R𝑛)
≤ 𝐶2−𝑛𝑗
2−𝑗𝛽
𝐼𝛽(𝜒𝐵𝑗
)𝐿𝑝2(⋅)(R𝑛)
≤ 𝐶2−𝑗𝛽
2−𝑛𝑗
𝜒𝐵𝑗
𝐿𝑝1(⋅)(R𝑛)
≤ 𝐶2−𝑗𝛽
𝜒𝐵𝑗
−1
𝐿𝑝
1(⋅)
(R𝑛).
(39)
Combining (37) and (39), we have the estimate
𝐼𝑚
𝛽,𝑏(𝑓𝑗) 𝜒𝑘
𝐿𝑝2(⋅)(R𝑛)≤ 𝐶(𝑘 − 𝑗 + 1)
𝑚
× ‖𝑏‖𝑚
BMO2(𝑘−𝑗)(𝛽−𝑛𝑟)
𝑓𝑗
𝐿𝑝1(⋅)(R𝑛).
(40)
Thus,
𝑉1≤ 𝐶‖𝑏‖
𝑚𝑞1
BMO
∞
∑
𝑘=−∞
(
𝑘−2
∑
𝑗=−∞
2𝛼𝑗𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)
×(𝑘 − 𝑗 + 1)𝑚
2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)
)
𝑞1
.
(41)
6 Journal of Function Spaces and Applications
In case of 1 < 𝑞1
< ∞, noting that 𝛽 − 𝑛𝑟 + 𝛼 < 0, byHölder’s inequality, we have
𝑉1≤ 𝐶‖𝑏‖
𝑚𝑞1
BMO
×
∞
∑
𝑘=−∞
(
𝑘−2
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞
1/2
)
× (
𝑘−2
∑
𝑗=−∞
(𝑘 − 𝑗 + 1)𝑚𝑞
2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞
1/2
)
𝑞1/𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
∞
∑
𝑘=𝑗+2
2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞
1/2
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO𝑓
𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(42)
In case of 0 < 𝑞1≤ 1, by inequality (21), we have
𝑉1≤ 𝐶‖𝑏‖
𝑚𝑞1
BMO
×
∞
∑
𝑘=−∞
𝑘−2
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
× (𝑘 − 𝑗 + 1)𝑚𝑞1
2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
×
∞
∑
𝑘=𝑗+2
(𝑘 − 𝑗 + 1)𝑚𝑞1
2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO𝑓
𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(43)
For 𝑉2, by Lemma 12(1), we obtain
𝑉2≤ 𝐶‖𝑏‖
𝑚𝑞1
BMO
∞
∑
𝑘=−∞
(
∞
∑
𝑗=𝑘−1
2𝛼𝑗𝑓𝑗
𝐿𝑝1(⋅)(R𝑛)2𝛼(𝑘−𝑗)
)
𝑞1
. (44)
If 1 < 𝑞1< ∞, by Hölder’s inequality, we have
𝑉2≤ 𝐶‖𝑏‖
𝑚𝑞1
BMO
∞
∑
𝑘=−∞
(
∞
∑
𝑗=𝑘−1
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
2𝛼(𝑘−𝑗)𝑞
1/2
)
× (
∞
∑
𝑗=𝑘−1
2𝛼(𝑘−𝑗)𝑞
1/2
)
𝑞1/𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
𝑗+1
∑
𝑘=−∞
2𝛼(𝑘−𝑗)𝑞
1/2
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO𝑓
𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(45)
If 0 < 𝑞1≤ 1, by inequality (21), we have
𝑉2≤ 𝐶‖𝑏‖
𝑚𝑞1
BMO
∞
∑
𝑘=−∞
∞
∑
𝑗=𝑘−1
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
2𝛼(𝑘−𝑗)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO
∞
∑
𝑗=−∞
2𝛼𝑗𝑞1
𝑓𝑗
𝑞1
𝐿𝑝1(⋅)(R𝑛)
𝑗+1
∑
𝑘=−∞
2𝛼(𝑘−𝑗)𝑞
1
≤ 𝐶‖𝑏‖𝑚𝑞1
BMO𝑓
𝑞1
�̇�𝛼,𝑞1
𝑝1(⋅)(R𝑛)
.
(46)
Combining the estimates for 𝑉1and 𝑉
2, consequently, we
have provedTheorem 7.
Acknowledgments
The authors thank the referees for their valuable commentsto the original version of this note. This paper is supportedby the NSF of China (no. 11201003); the Natural ScienceFoundation of Anhui Higher Education Institutions of China(no. KJ2011A138; no. KJ2013B034).
References
[1] D. R. Adams, “A note on Riesz potentials,” Duke MathematicalJournal, vol. 42, no. 4, pp. 765–778, 1975.
[2] S. Lu and D. Yang, “Hardy-Littlewood-Sobolev theorems offractional integration on Herz-type spaces and its applications,”Canadian Journal of Mathematics, vol. 48, no. 2, pp. 363–380,1996.
[3] S. G. Shi and Z. W. Fu, “Boundedness of sublinear operatorswith rough kernels on weighted Morrey spaces,” Journal ofFunction Spaces and Applications, vol. 2013, Article ID 784983,9 pages, 2013.
[4] O. Kováčik and J. Rákosnı́k, “On spaces 𝐿𝑝(𝑥) and 𝑊𝑘,𝑝(𝑥),”Czechoslovak Mathematical Journal, vol. 41, no. 4, pp. 592–618,1991.
[5] L. Diening and M. Růžička, “Calderón-Zygmund operators ongeneralized Lebesgue spaces 𝐿𝑝(⋅) and problems related to fluiddynamics,” Journal für die Reine und Angewandte Mathematik,vol. 563, pp. 197–220, 2003.
[6] C. Capone, D. Cruz-Uribe, and A. Fiorenza, “The fractionalmaximal operator and fractional integrals on variable 𝐿𝑝spaces,” Revista Mathemática Iberoamericana, vol. 23, no. 3, pp.743–770, 2007.
[7] M. Izuki, “Commutators of fractional integrals on Lebesgueand Herz spaces with variable exponent,” Rendiconti del CircoloMatematico di Palermo, vol. 59, no. 3, pp. 461–472, 2010.
[8] L. Diening, “Riesz potential and Sobolev embeddings on gene-ralized Lebesgue and Sobolev spaces 𝐿𝑝(⋅) and 𝑊𝑘,𝑝(⋅),” Mathe-matische Nachrichten, vol. 268, pp. 31–43, 2004.
[9] S. Z. Lu, D. C. Yang, and G. E. Hu, Herz Type Spaces and TheirApplications, Science Press, Beijing, China, 2008.
[10] Z.-W. Fu, Z.-G. Liu, S.-Z. Lu, andH.-B.Wang, “Characterizationfor commutators of 𝑛-dimensional fractional Hardy operators,”Science in China A, vol. 50, no. 10, pp. 1418–1426, 2007.
[11] S. Lu, Q. Wu, and D. Yang, “Boundedness of commutators onHardy type spaces,” Science in China A, vol. 45, no. 8, pp. 984–997, 2002.
Journal of Function Spaces and Applications 7
[12] T. Zhou, Commutators of Fractional Integrals on Spaces WithVariable Exponent, Dalian Maritime University, 2012.
[13] H. B.Wang, Z.W. Fu, andZ.G. Liu, “Higher order commutatorsof Marcinkiewicz integrals on variable Lebesgue spaces,” ActaMathematica Scientia A, vol. 32, no. 6, pp. 1092–1101, 2012.
[14] H. B.Wang, Function Spaces withVariable Exponent andRelatedTopics, China University of Mining and Technology, 2012.
[15] A. Nekvinda, “Hardy-Littlewood maximal operator on𝐿𝑝(𝑥)
(R𝑛),” Mathematical Inequalities & Applications, vol. 7, no.2, pp. 255–265, 2004.
[16] M. Izuki, “Boundedness of commutators on Herz spaceswith variable exponent,” Rendiconti del Circolo Matematico diPalermo, vol. 59, no. 2, pp. 199–213, 2010.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of