research article higher order commutators of fractional ...usual lebesgue spaces 1 (r ) to 2 (r )...

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; wangliwei8013@163.com

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.

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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


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


≤ ∫𝑅𝑗

𝑏 (𝑥) − 𝑏 (𝑦)𝑚


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




1−𝑛)

‖𝑏‖𝑚

Lip𝛽1

𝑓𝑗


×𝜒𝐵𝑘


𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)


1)

‖𝑏‖𝑚

Lip𝛽1

𝑓𝑗


×𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)

𝜒𝐵𝑘

−1

𝐿𝑝

2(⋅)

(R𝑛)


1)

‖𝑏‖𝑚

Lip𝛽1

𝑓𝑗


×𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)

𝜒𝐵𝑗

−1

𝐿𝑝

2(⋅)

(R𝑛)

𝜒𝐵𝑗

𝐿𝑝

2(⋅)

(R𝑛)𝜒𝐵𝑘

𝐿𝑝

2(⋅)

(R𝑛)


1)

2𝑛𝑟(𝑗−𝑘)

‖𝑏‖𝑚

Lip𝛽1

𝑓𝑗


×𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)

𝜒𝐵𝑗

−1

𝐿𝑝

2(⋅)

(R𝑛).

(24)


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


≤ 𝐶2−𝑛𝑗

2−𝑗(𝛽+𝑚𝛽

1)𝐼𝛽+𝑚𝛽

1

(𝜒𝐵𝑗

)𝐿𝑝2(⋅)(R𝑛)

≤ 𝐶2−𝑗(𝛽+𝑚𝛽

1)

2−𝑛𝑗

𝜒𝐵𝑗


≤ 𝐶2−𝑗(𝛽+𝑚𝛽

1)𝜒𝐵𝑗

−1

𝐿𝑝

1(⋅)

(R𝑛).

(26)

Combining (24) and (26), we have the estimate

𝐼𝑚


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


2(𝑘−𝑗)(𝛽+𝑚𝛽

1−𝑛𝑟+𝛼)𝑞

1/2

)

× (

𝑘−2

∑

𝑗=−∞



1/2

)

𝑞1/𝑞

1

≤ 𝐶‖𝑏‖𝑚𝑞1

Lip𝛽1

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


∞

∑

𝑘=𝑗+2



1/2


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




1


Lip𝛽1

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


∞

∑

𝑘=𝑗+2



1


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


2𝛼(𝑘−𝑗)𝑞

1/2

)

× (

∞

∑

𝑗=𝑘−1


1/2

)

𝑞1/𝑞

1


Lip𝛽1

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


𝑗+1

∑

𝑘=−∞


1/2


Lip𝛽1

𝑓𝑞1

�̇�𝛼,𝑞1

𝑝1(⋅)(R𝑛)

.

(32)


𝑈2≤ 𝐶‖𝑏‖

𝑚𝑞1

Lip𝛽1

∞

∑

𝑘=−∞

∞

∑

𝑗=𝑘−1

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1



1


Lip𝛽1

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


𝑗+1

∑

𝑘=−∞


1


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)


By inequality (21), we obtain

𝐼𝑚

𝛽,𝑏(𝑓)

𝑞1

�̇�𝛼,𝑞2

𝑝2(⋅)(R𝑛)

= 𝐶(

∞

∑

𝑘=−∞

2𝛼𝑞2𝑘𝐼𝑚

𝛽,𝑏(𝑓) 𝜒𝑘

𝑞2


)

𝑞1/𝑞2

≤ 𝐶

∞

∑

𝑘=−∞

2𝛼𝑞1𝑘

(

𝑘−2

∑

𝑗=−∞

𝐼𝑚


𝐿𝑝2(⋅)(R𝑛))

𝑞1

+ 𝐶

∞

∑

𝑘=−∞

2𝛼𝑞1𝑘

(

∞

∑

𝑗=𝑘−1

𝐼𝑚


𝐿𝑝2(⋅)(R𝑛))

𝑞1

= 𝑉1+ 𝑉2.

(35)

For 𝑉1, using Hölder’s inequality and Lemma 8, we have

𝐼𝑚



∫𝑅𝑗

𝑏 (𝑥) − 𝑏 (𝑦)𝑚

𝑓𝑗(𝑦)

𝑑𝑦 ⋅ 𝜒

𝑘(𝑥)


𝑚

∑

𝑖=0

𝐶𝑖

𝑚

𝑏 (𝑥) − 𝑏

𝐵𝑗

𝑚−𝑖

× ∫𝑅𝑗

𝑏𝐵𝑗

− 𝑏 (𝑦)

𝑖 𝑓𝑗(𝑦)

𝑑𝑦


𝑓𝑗


×

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




𝑓𝑗


×

𝑚

∑

𝑖=0

𝐶𝑖

𝑚

(𝑏 (𝑥) − 𝑏

𝐵𝑗

)𝑚−𝑖

𝜒𝑘


×

(𝑏𝐵𝑗

− 𝑏)𝑖

𝜒𝑗

𝐿𝑝

1(⋅)

(R𝑛)


𝑓𝑗


×

𝑚

∑

𝑖=0

𝐶𝑖

𝑚(𝑘 − 𝑗)

𝑚−𝑖

‖𝑏‖𝑚−𝑖

BMO𝜒𝐵𝑘


× ‖𝑏‖𝑖

BMO𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)

= 𝐶(𝑘 − 𝑗 + 1)𝑚

‖𝑏‖𝑚

BMO𝑓𝑗


× 2𝑘(𝛽−𝑛)

𝜒𝐵𝑘


𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)

≤ 𝐶(𝑘 − 𝑗 + 1)𝑚

‖𝑏‖𝑚

BMO𝑓𝑗


× 2𝑘𝛽𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)

𝜒𝐵𝑘

−1

𝐿𝑝

2(⋅)

(R𝑛)

≤ 𝐶(𝑘 − 𝑗 + 1)𝑚

‖𝑏‖𝑚

BMO𝑓𝑗


× 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


× 2𝑘𝛽

2𝑛𝑟(𝑗−𝑘)

𝜒𝐵𝑗

𝐿𝑝

1(⋅)

(R𝑛)

𝜒𝐵𝑗

−1

𝐿𝑝

2(⋅)

(R𝑛).

(37)

Note that

𝐼𝛽(𝜒𝐵𝑗

) (𝑥) ≥ ∫𝐵𝑗

𝑑𝑦


⋅ 𝜒𝐵𝑗

(𝑥) ≥ 𝐶2𝑗𝛽

⋅ 𝜒𝐵𝑗

(𝑥) .

(38)

By Lemmas 8 and 11, we obtain

𝜒𝐵𝑗

−1

𝐿𝑝

2(⋅)

(R𝑛)≤ 𝐶2−𝑛𝑗

𝜒𝐵𝑗


≤ 𝐶2−𝑛𝑗

2−𝑗𝛽

𝐼𝛽(𝜒𝐵𝑗

)𝐿𝑝2(⋅)(R𝑛)

≤ 𝐶2−𝑗𝛽

2−𝑛𝑗

𝜒𝐵𝑗


≤ 𝐶2−𝑗𝛽

𝜒𝐵𝑗

−1

𝐿𝑝

1(⋅)

(R𝑛).

(39)

Combining (37) and (39), we have the estimate

𝐼𝑚


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


×(𝑘 − 𝑗 + 1)𝑚

2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)

)

𝑞1

.

(41)


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


2(𝑘−𝑗)(𝛽−𝑛𝑟+𝛼)𝑞

1/2

)

× (

𝑘−2

∑

𝑗=−∞

(𝑘 − 𝑗 + 1)𝑚𝑞


1/2

)

𝑞1/𝑞

1


BMO

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


∞

∑

𝑘=𝑗+2


1/2


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


× (𝑘 − 𝑗 + 1)𝑚𝑞1


1


BMO

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


×

∞

∑

𝑘=𝑗+2

(𝑘 − 𝑗 + 1)𝑚𝑞1


1


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



1/2

)

× (

∞

∑

𝑗=𝑘−1


1/2

)

𝑞1/𝑞

1


BMO

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


𝑗+1

∑

𝑘=−∞


1/2


BMO𝑓

𝑞1

�̇�𝛼,𝑞1

𝑝1(⋅)(R𝑛)

.

(45)


𝑉2≤ 𝐶‖𝑏‖

𝑚𝑞1

BMO

∞

∑

𝑘=−∞

∞

∑

𝑗=𝑘−1

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1



1


BMO

∞

∑

𝑗=−∞

2𝛼𝑗𝑞1

𝑓𝑗

𝑞1


𝑗+1

∑

𝑘=−∞


1


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

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[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.

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[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.

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(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.

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