Applied Mathematics, 2016, 7, 1165-1182 Published Online June 2016 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2016.710104

How to cite this paper: Abdalmonem, A., Abdalrhman, O. and Tao, S.P. (2016) Boundedness of Fractional Integral with Va-riable Kernel and Their Commutators on Variable Exponent Herz Spaces. Applied Mathematics, 7, 1165-1182. http://dx.doi.org/10.4236/am.2016.710104

Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz Spaces Afif Abdalmonem1,2, Omer Abdalrhman1,3, Shuangping Tao1* 1College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China 2Faculty of Science, University of Dalanj, Dalanj, Sudan 3College of Education, Shendi University, Shendi, Sudan

Received 25 April 2016; accepted 26 June 2016; published 29 June 2016

Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents ( ) ( )p q,⋅ ⋅ . By using the properties of the variable ex-ponents Lebesgue spaces, the boundedness of the fractional integral operator and their commu-tator generated by Lipschitz function is obtained on those Herz spaces.

Keywords Fractional Integral, Variable Kernel, Commutator, Variable Exponent, Lipschitz Space, Herz Spaces

1. Introduction Let 0 nµ< < , ( ) ( ) ( )1 1n r nL L S r∞ −Ω∈ × ≥ is homogenous of degree zero on n , 1nS − denotes the unit sphere in n . If

(i) For any , nx z∈ , one has ( ) ( ), ,x z x zλΩ = Ω ;

(ii) ( ) ( ) ( ) ( )( )1 11

: sup , dn r n nn

r rL L S s

xx z zσ∞ − −×

∈

′ ′Ω = Ω < ∞∫

The fractional integral operator with variable kernel ,T µΩ is defined by

*Corresponding author.

http://www.scirp.org/journal/amhttp://dx.doi.org/10.4236/am.2016.710104http://dx.doi.org/10.4236/am.2016.710104http://www.scirp.orghttp://creativecommons.org/licenses/by/4.0/

A. Abdalmonem et al.

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( ) ( ) ( ),,

d ,n nx x y

T f x f y yx y

µ µΩ −

Ω −=

−∫

The commutators of the fractional integral is defined by

( ) ( ) ( ) ( )( ) ( ),,

, d ,nmm

n

x x yb T f x b x b y f y y

x yµ µΩ −

Ω − = − −∫

When 1µ ≡ , the above integral takes the Cauchy principal value. At this time 0µ ≡ , ,T µΩ is much more close related to the elliptic partial equations of the second order with variable coefficients. Now we need the further assumption for ( ),x zΩ . It satisfies

( ) ( )1 , d 0,n ns x z z xσ− ′ ′Ω = ∀ ∈∫

For 1r ≥ , we say Kernel function ( ),x zΩ satisfies the rL -Dini condition, if Ω meets the conditions (i), (ii) and

( )10

drω δ

δδ

< ∞∫

where ( )rω δ denotes the integral modulus of continuity of order r of Ω defined by

( ) ( ) ( ) ( )( )11

,sup , , dn

n

r rr s

xx z x z z

ρ δω δ ρ σ−

∈ <

′ ′ ′= Ω −Ω∫

where ρ is the a rotation in n

1sup

nz Sz zρ ρ

−′∈

′ ′= −

when 1Ω ≡ , ,T µΩ is the fraction integral operator

( ) ( ), d .n nf y

T f x yx y

µ µΩ −= −∫

The corresponding fractional maximal operator with variable kernel is defined by

( ) ( ),0

1sup , d .n x y rrM f x x x y y

rµ µΩ − − = Ω −∫

We can easily find that when 1Ω ≡ ,M µΩ is just the fractional maximal operator

( ) ( ),0

1sup dn x y rrM f x f y y

rµ µΩ − − = ∫

Especially, in the case 0µ ≡ , the fractional maximal operator reduces the Hardy-Littelewood maximal operator.

Many classical results about the fractional integral operator with variable kernel have been achieved [1]-[5]. In 1971, Muckenhoupt and Wheeden [6] had proved the operator ,T µΩ was bounded from

pL to qL . In 1991, Kováčik and Rákosník [7] introduced variable exponents Lebesgue and Sobolev spaces as a new method for dealing with nonlinear Dirichet boundary value problem. In the last 20 years, more and more researchers have been interested in the theory of the variable exponent function space and its applications [8]-[14]. In 2012, Wu Huiling and Lan Jiacheng [15] proved the bonudedness property of ,T µΩ with a rough kernel on variable exponents Lebesgue spaces.

Recently, Wang and Tao [16] introduced the class of Herz spaces with two variable exponents, and also studied the Parameterized Littlewood-Paley operators and their commutators on Herz spaces with variable exponents.

The main purpose of this paper is to discuss the boundedness of the fractional integral with variable kernel

,T µΩ and their commutators ,,mb T µΩ are bonuded on Herz spaces with two variable exponents or not.

A. Abdalmonem et al.

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Throughout this paper E denotes the Lebesgue measure, Eχ means he characteristic function of a measurable set nS ⊂ . C always means a positive constant independent of the main parameters and may change from one occurrence to another.

2. Definition of Function Spaces with Variable Exponent In this section we define the Lebesgue spaces with variable exponent and Herz spaces with two variable ex- ponent, and also define the mixed Lebesgue sequence spaces.

Let E be a measurable set in n with 0E > . We first define the Lebesgue spaces with variable exponent. Definition 2.1. see [1] Let ( ) [ ): 1,p E⋅ → ∞ be a measurable function. The Lebesgue space with variable

exponent ( ) ( )pL E⋅ is defined by

( ) ( ) ( )( )

is measurable : d for some constant 0p x

p

E

f xL E f x η

η⋅

= < ∞ > ∫

The space ( ) ( )plocL E⋅ is defined by ( ) ( ) ( ) ( ){ }is measurable : for all compactp plocL E f f L K K E⋅ ⋅= ∈ ⊂

The Lebesgue spaces ( ) ( )pL E⋅ is a Banach spaces with the norm defined by

( )( )( ) ( )

inf 0 : d 1pp x

L E E

f xf xη

η⋅

= > ≤ ∫

We denote

( ){ }essinf : ,p p x x E− = ∈ ( ){ }esssup :p p x x E+ = ∈ . Then ( )E consists of all ( )p ⋅ satisfying 1p− > and p+ < ∞ . Let M be the Hardy-Littlewood maximal operator. We denote ( )EB to be the set of all function ( ) ( )p E⋅ ∈ satisfying the M is bounded on ( ) ( )pL E⋅ . Definition 2.2. see [17] Let ( ) ( ) ( ),p q E⋅ ⋅ ∈ . The mixed Lebesgue sequence space with variable exponent ( ) ( )( )q pl L⋅ ⋅ is the collection of all sequences { } 0j jf

∞

= of the measurable functions on n such that

{ }( ) ( )( ) ( ) ( )( )0 0

inf 0 : 1q pq pj

j j l Ll L j

ff Qη

µ⋅ ⋅⋅ ⋅

∞∞

==

= > ≤ < ∞

( ) ( )( ) { }( )( )

( )

( )

100inf ; d 1nq p

p x

jj j Rjl L j q x

j

f xQ f xµ

µ⋅ ⋅

∞∞

==

= ≤

∑ ∫

Noticing q+ < ∞ , we see that

( ) ( )( ) { }( )( )

( )( )0 0

pq p q

qj jjl L Lj

Q f f ⋅⋅ ⋅ ⋅∞∞ ⋅

==

= ∑

Let { } 1: 2 , \ , ,n kk k k k k CkB x x C B B kχ χ−= ∈ ≤ = = ∈ Definition 2.3. see [16] Let ( ) ( ) ( ), , np qα ∈ ⋅ ⋅ ∈ . The homogeneous Herz space with variable ex-

ponent ( )( ) ( ),q npKα ⋅⋅ is defined by

( )( ) ( ) ( ) { }( )

( )( )( ){ },, \ 0 : q npq pn nLocp KK f L f αα ⋅⋅⋅ ⋅⋅ = ∈ < ∞

where

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1168

( )( )( ) { } ( ) ( )( )

( )

( )( )

,0

22 inf 0 : 1q n

pq ppq

qkkk

kK k l L kL

ff fα

αα χχ η

η⋅

⋅⋅ ⋅⋅⋅

⋅∞∞

==−∞

= = > ≤

∑

Remark 2.1. see [16] (1) If ( ) ( ) ( )1 2, nq q⋅ ⋅ ∈ satisfying ( ) ( )1 2q q+ +≤ , then ( )

( ) ( ) ( ) ( ) ( )1 2, ,q qn np pK Kα α⋅ ⋅⋅ ⋅⊂

(2) If ( ) ( ) ( )1 2, nq q⋅ ⋅ ∈ and ( ) ( )1 2q q+ +≤ , then ( )( ) ( )

2

1

nqq

⋅∈

⋅ and ( )

( )2

1

1qq

⋅≥

⋅. Thus, by Lemma 3.7

and Remark 2.2, for any ( )( ) ( ),q npf Kα ⋅⋅∈ , we have

( )

( )( )

( )

( )( )

( )( )

*2 1 1

12 1

( )2 2 2

1h h

pp pqq q

pp pq q qk k kk k k

k k kLL L

f f fα α αχ χ χη η η ⋅⋅ ⋅

⋅⋅ ⋅

⋅ ⋅ ⋅∞ ∞ ∞

=−∞ =−∞ =−∞

≤ ≤ ≤

∑ ∑ ∑

where

( )( )( )( )

2

1

2

1

2, 1

2, 1

kk

h kk

fqq

pfq

q

α

α

χη

χη

−

+

⋅≤ ⋅

= ⋅ > ⋅

0*

0

min , 1

max , 1

h hh h

h hh h

p ap

p a

∞

∈ =

∞

∈ =

≤= >

∑

∑

This implies that ( )( ) ( ) ( ) ( ) ( )1 2, ,q qn np pK Kα α⋅ ⋅⋅ ⋅⊂ .

Remark 2.2. Let , 0,1h hh a p∈ ≥ ≤ < ∞ . then *

0 0

p

h hh h

a a∞ ∞

= =

≤

∑ ∑

where

0*

0

min , 1

max , 1

h hh h

h hh h

p ap

p a

∞

∈ =

∞

∈ =

≤= >

∑

∑

Definition 2.4. see [18] For 0 1β< ≤ , the Lipschitz space ( )nLipβ is defined by

( ) ( ) ( ), ,

: supn

nLip

x y x y

f x f yLip f f

x yββ β

∈ ≠

− = = < ∞ −

(1.1)

3. Properties of Variable Exponent In this section we state some properties of variable exponent belonging to the class ( )nB and

( ) ( )1n r nL L S∞ −Ω∈ × . Proposition 3.1. see [1] If ( ) ( )np ⋅ ∈ satisfies

( ) ( ) ( ) , 1 2logCp x p y x yx y−

− ≤ − ≤−

A. Abdalmonem et al.

1169

( ) ( ) ( ) ,logCp x p y y xe x

− ≤ ≥+

then, we have ( ) ( )np ⋅ ∈ B . Proposition 3.2. see [15] Suppose that ( ) ( )1 np ⋅ ∈ B , ( ) ( )1n r nL L S∞ −Ω∈ × . Let ( )1

0 np

µ+

< ≤ , and

define the variable exponent ( )2p ⋅ by: ( ) ( )1 21 1 =

p x p x nµ

− . Then we have that for all ( ) ( )1p nf L ⋅∈ ,

( )( ) ( )( )12, p np n LLT f c fµ ⋅⋅Ω ≤

Proposition 3.3. Suppose that ( ) ( )1 np ⋅ ∈ B , ( )nb Lipβ∈ , 0 1β< ≤ , ( ) ( )1n r nL L S∞ −Ω∈ × . Let

( )10 nm

pµ β

+

< + ≤ , and define the variable exponent ( )2p ⋅ by: ( ) ( )1 21 1 m

p x p x nµ β+

− = . Then

( )( ) ( )( )12,, p np n

mmLip LL

b T f C b fβµ

⋅⋅Ω ≤

Proof

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )

( )

,

,

,, d

,d

,d

n

n

n

mmn

m

n

n mLip

mLip

x x yb T f x b x b y f y y

x y

x x yb x b y f y y

x y

x x yC b f y y

x y

C b T f

β

β

µ µ

µ

µ β

µ β

Ω −

−

− −

Ω +

Ω − = − −

Ω −≤ −

−

Ω −≤

−

≤

∫

∫

∫

By Proposition 3.2, we get

( )( ) ( ) ( ) ( )( )12 2, ,, .p np pnmm

mLip Lip LL Lb T f C b T f C b f

β βµ µ β⋅⋅ ⋅Ω Ω +

≤ ≤

Now, we need recall some lemmas Lemma 3.1. see [13] Given ( ) [ ): 1,np ⋅ → ∞ have that for all function f and g,

( ) ( ) ( )( ) ( )( )d p pn nn L Lf x g x x C f g ′⋅ ⋅≤∫

Lemma 3.2. see [19] Suppose that 0 nµ< < , 1r > , ( ) ( )1n r nL L S∞ −Ω∈ × satisfies the rL -Dini con- dition. If there exists an 00 1 2α< < such that 0y Rα< then

( ) ( ) ( )1

22

, ,d d

r r n y Rnrr

n nR x Ry R

yx x y x xx CR

Rx y x

µ

µ µ

ω δδ

δ

− +

− −< <

Ω − Ω − ≤ + −

∫ ∫

Lemma 3.3. see [20] Suppose that nx∈ , the variable function ( )q x is defined by ( ) ( )1 1 1

p x q q x= +

,

then for all measurable function f and g, we have

( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )p qn q n nL L Lf x g x C g x f x⋅ ⋅≤

Lemma 3.4. see [21] Suppose that ( ) ( )np ⋅ ∈ B and 0 p p− +< ≤ < ∞ .

A. Abdalmonem et al.

1170

1) For any cube and 2nQ ≤ , all the Qχ ∈ , then: ( )( )1

pp x

Q LQχ ⋅ ≈

2) For any cube and 1Q ≥ , then ( )1

pp

Q LQχ ∞⋅ ≈ where

( )limxp p x∞ →∞=

Lemma 3.5. see [22] If ( ) ( )np ⋅ ∈ B , then there exist constants 1 2, , 0Cδ δ > such that for all balls B in n

and all measurable subset S R⊂

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

11

1

21

1

, , p pn n

pp nn

p n

p n

SB L L

S B LL

S L

B L

B SC C

S B

SC

B

δ

δ

χχ

χ χ

χ

χ

′⋅ ⋅

′ ⋅⋅

⋅

⋅

≤ ≤

≤

Lemma 3.6. see [13] If ( ) ( )np ⋅ ∈ B , there exist a constant 0C > such that for any balls B in n . we have

( )( ) ( )( )1

p pn nB BL L CBχ χ ′⋅ ⋅ ≤

Lemma 3.7. see [16] Let ( ) ( ) ( ), np q⋅ ⋅ ∈ . If ( ) ( )p qf L ⋅ ⋅∈ , then ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( )( )min , max , p q p q p q p qpq q q q qL L L LLf f f f f+ − + −⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅

⋅≤ ≤

4. Main Theorems and Their Proof Theorem 1. Suppose that 0 nµ< < , 2 1n nµ δ α δ− < < , ( ) ( ) ( )1 2n r nL L S r p∞ − +Ω∈ × > ,

( ) ( ) ( )1 2, nq q⋅ ⋅ ∈ with ( ) ( )2 1q q− +≥ . And let ( ) ( )1 np ⋅ ∈ B satisfy ( )10 n

pµ

+

< ≤ and define the vari-

able exponent ( )2p x by ( ) ( )1 21 1

p x p x nµ

− = . Then the operators ,T µΩ is bounded from ( )( ) ( )11,q npKα ⋅⋅ to

( )( ) ( )22,q npKα ⋅⋅ .

Theorem 2. Let ( ) , .nb Lip mβ∈ ∈ Suppose that ( ) 2 10 , n m n nµ µ β δ α δ< < + − < < , ( ) ( ) ( )1 2n r nL L S r p∞ − +Ω∈ × > , ( ) ( ) ( )1 2, nq q⋅ ⋅ ∈ with ( ) ( )2 1q q− +≥ . If ( ) ( )1 np ⋅ ∈ B satisfy

( )10 nm

pµ β

+

< + ≤ and define the variable exponent ( )2p x by ( ) ( )1 21 1 m

p x p x nµ β+

− = . Then the com-

mutators ,,mb T µΩ is bounded from ( )

( ) ( )11,q npKα ⋅⋅ to ( )( ) ( )22,q npKα ⋅⋅ . Proof of Theorem1: Let ( )

( ) ( )11,q npf Kα ⋅⋅∈ . We write

( ) ( ) ( )j jj j

f x f x f xχ∞ ∞

=−∞ =−∞

= =∑ ∑

From definition of ( )( ) ( ),q npKα ⋅⋅

( )( )( )

( )( )

( )( )

2

, 2222

,,

2, inf 0 : 1q n

ppq

qkk

Kk

L

T fT f α

αµ

µ

χη

η⋅ ⋅⋅⋅

⋅∞

ΩΩ

=−∞

= > ≤

∑

Since

A. Abdalmonem et al.

1171

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )

2

22

2 2

2 22 2

2

22

,

2

, ,

11 12 13 11

1

, ,1 2

12 13

2

2 2

2 2

pq

p pq q

pq

qkk

L

q qk

k kj k j k

j j

L L

qk

k kj k j k

j k j k

L

T f

T f T f

T f T f

αµ

α αµ µ

α αµ µ

χη

χ χ

η η η η

χ χ

η η

⋅⋅

⋅ ⋅⋅ ⋅

⋅⋅

⋅

Ω

⋅ ⋅∞ −

Ω Ω=−∞ =−∞

⋅+ ∞

Ω Ω= − = +

≤ ≤ + +

+ +

∑ ∑

∑ ∑( )

( )( )

2

22

pq

q

L

⋅⋅

⋅

where

( )( ) ( )( )2 2

2

11 ,2q p

kk

j kj k l L

T fα µη χ⋅ ⋅

∞−

Ω=−∞ =−∞

=

∑

( )( ) ( )( )2 2

1

12 ,1

2q p

kk

j kj k k l L

T fα µη χ⋅ ⋅

∞+

Ω= − =−∞

=

∑

( )( ) ( )( )2 2

13 ,2

2q p

kj k

j k k l L

T fα µη χ⋅ ⋅

∞∞

Ω= + =−∞

=

∑

And 11 12 13η η η η= + + , thus

( ) ( )

( )( )

2

22

,2

pq

qk

j k

k

L

T fC

αµ χ

η ⋅⋅

⋅∞ Ω

=−∞

≤

∑

That is

( )( )( )( ) [ ], 22, 11 12 13

.q npK

T f C Cαµ η η η η⋅⋅

Ω ≤ = + +

This implies only to prove ( )( )( ), 1111 12 13

, , q npK

C f αη η η ⋅⋅

≤

. Denote ( )( )( ), 111

q npK

f αη ⋅⋅

=

Now we consider 12η . Applying Lemma 3.7

( )( )

( )( )

( )

( )

( )

( )( )

( )

12 2

22

2

12

2

1 1

, ,1 1

1 1

1 ,

1 1

2 2

2

pp

q

p

q q kk k

k kj k j k

j k j k

k k

LL

q kkk j k

k j kL

T f T fC

T fC

α αµ µ

αµ

χ χ

η η

χ

η

⋅⋅

⋅

⋅

⋅+ +

Ω Ω∞ ∞= − = −

=−∞ =−∞

∞ + Ω

=−∞ = −

≤

≤

∑ ∑∑ ∑

∑ ∑

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1172

where

( )

( )( )

( )

( )( )

( )( )

( )

( )( )

2

22

2

22

1

,1

21

12

1

,1

21

2, 1

2, 1

pq

pq

qk

kj k

j k

L

qk

kj k

j k

L

T fq

q k

T fq

αµ

αµ

χ

η

χ

η

⋅⋅

⋅⋅

⋅+

Ω= −

−

⋅+

Ω= −

+

≤

= >

∑

∑

By the Proposition 3.2, we get

( )( )

( )( )

( )

( )2

12

1

22

1

, 11

11 1

22

p

pq

qk

k q kkj k kj k j

k k j kL

L

T ff

C

ααµ χ

η η⋅

⋅⋅

⋅+

Ω∞ ∞ += −

=−∞ =−∞ = −

≤

∑∑ ∑ ∑

Since ( )( ) ( )11,q npf Kα ⋅⋅∈ , then we have

( )1

21

p

jj

L

fα χη

⋅

≤ , and

( )

( )( )

1

11

1

21

pq

qjj

jL

fα χη ⋅

⋅

⋅∞

=−∞

≤

∑

By Lemma 3.7 and Remark 2.2, we get

( )( )

( )( )

( )

( )

( )

( )( )

( )

( )

( )

2 12

1 1

1

122

*

1

1

1

1

,1

1 1

1

22

2

p

qpq

p

q

qk q k

kq qj k k

j k k

k k L

LL

q

qkk

k L

L

T ff

C

fC C

αµ α

α

χχ

η η

χη

+

⋅

⋅⋅⋅

⋅

⋅

⋅+

⋅Ω∞ ∞= −

=−∞ =−∞

⋅∞

=−∞

≤

≤ ≤

∑∑ ∑

∑

Hence ( ) ( ) ( )11 2 2,p p q k+ − ≤ , and ( )( )

12

*1

mink

q kq

q∈+

=

, this implies that

( )( )( ), 1112 1

q npK

C C f αη η ⋅⋅

≤ ≤

Now, we estimate of 11η using size condition of jf and Minkowski inequality, when 1j k≤ − we get,

( ) ( )( ) ( )( ) ( )

( )( )2

2

,, ,

dp n jp n

j k j kn nBLL

x x y x xT f f y y

x y xµ µ µχ χ⋅

⋅

Ω − −

Ω − Ω≤ −

−∫

A. Abdalmonem et al.

1173

Since 2r p+> we define the variable exponent

( ) ( )2 21 1 1

p x r p x= +

, by Lemma 3.3 we get

( ) ( )( )( )

( ) ( )

( )( )( )

( ) ( )

( )( )( )

2

2

2

, , , ,

, ,

p x n

r np n

p x nkr n

k kn n n n L

LL

Bn n LL

x x y x x x x y x xx y x x y x

x x y x xx y x

µ µ µ µ

µ µ

χ χ

χ

⋅

− − − −

− −

Ω − Ω Ω − Ω− ≤ −

− −

Ω − Ω≤ −

−

According Lemma 3.4 and the formula ( ) ( )2 21 1 1

p x p x r= −

, then we have

( )( ) ( )( ) ( )( )2 2 11 1

p x p x p xn n nk k kr r nB B k B kL L L

B Bµ

χ χ χ− −

−≈ ≈

(1.2)

By Lemma 3.2, we get

( ) ( )

( )

( )

( ) ( )

( )

12

12

1

0

1

, ,d

2

2 1 d

2

k

kr n

n ynrr

n n kyL

n n j k rr

nk nr

yx x y x xCR

x y x

CR

C

µ

µ µ

µβ

µ

ω δδ

δ

ω δδ

δ

− − +

− − −

− + −

− − +

Ω − Ω − ≤ + −

≤ +

≤

∫

∫

It follows that

( ) ( )( ) ( ) ( )( )12, 2 d p x np n kjkn

j k j BB LLT f C f y yµ χ χ⋅

−Ω ≤ ∫

(1.3)

By the Equation (1.3) and using Lemmas 3.1, 3.5, 3.6, 3.7, we can obtain

( )( )

( )( )

( )

( )

( )

( )( )

( )

( )( )( )

( )

22 2

22

2

2 22 2

1 ( )111 1

2 2

, ,

1 1

2 2

1 1

2 2

2 2 2 2

pp

q

p ppj kpn n

q q kk k

k kj k j k

j j

k k

LL

q k q kk k

j jk kn k knk B BL LLk j k jL L

T f T fC

f fC C

α αµ µ

α α

χ χ

η η

χ χ χη η

⋅⋅

⋅

⋅ ⋅′ ⋅⋅

⋅− −

Ω Ω∞ ∞=−∞ =−∞

=−∞ =−∞

∞ − ∞ −− −

=−∞ =−∞ =−∞ =−∞

≤

≤ ≤

≤

∑ ∑∑ ∑

∑ ∑ ∑ ∑

( )( )

( )

( )

( )( )

( )( )

( )

( )( )( )

( ) ( )( )

( )( )

2 22 2

1 1

1 11

22

1 1

1

1 1

2 2

1 1

1

2

1

2 2 2

22

pj

p pn npk

p q n

q k q kk kBj jj k nk kL

k j k jBL LL

q k

q qjk j kj k n

k jL

f fC C

fC

δα α

αδ α

χ

η ηχ

χ

η

′ ⋅

⋅ ⋅′ ⋅

+

⋅ ⋅

∞ − ∞ −−

=−∞ =−∞ =−∞ =−∞

⋅∞ −

− −

=−∞ =−∞

≤

≤

∑ ∑ ∑ ∑

∑ ∑

where

A. Abdalmonem et al.

1174

( )

( )( )

( )

( )( )

( )( )

( )

( )( )

2

22

2

22

2

,

21

22

2

,

21

2, 1

2, 1

pq

pq

qk

kj k

j

L

qk

kj k

j

L

T fq

q k

T fq

αµ

αµ

χ

η

χ

η

⋅⋅

⋅⋅

⋅−

Ω=−∞

−

⋅−

Ω=−∞

+

≤

= >

∑

∑

Since ( )( ) ( )11,q npf Kα ⋅⋅∈ , then we have

( )1

21

p

jj

L

fα χη

⋅

≤ , and

( )

( )( )

1

11

1

21

pq

qjj

jL

fα χη ⋅

⋅

⋅∞

=−∞

≤

∑

Now if ( )1 1q + < , then we have

( )( )

( )( )

( )( )( )

( ) ( )( )

( )( )

( )

( ) ( )( )

( )( )

222

11

1

1 122

*

1

1

1 1

2

, 2

1 1

21

2 22

22

p q npq

p q n

q kqk

qk qjj k k j kj j k n

k k jL

Lq

qjj k j k n

j k jL

T f fC

fC C

ααµ

δ α

αδ α

χχ

η η

χ

η

+

⋅ ⋅⋅⋅

⋅ ⋅

⋅−

⋅Ω∞ ∞ −

=−∞ − −

=−∞ =−∞ =−∞

⋅∞ ∞

− −

=−∞ = +

≤

≤ ≤

∑∑ ∑ ∑

∑ ∑

where ( )( )

22

*1

mink

q kq

q∈+

=

If ( )1 1q + ≥ , then we have

( )( )

( )( )

( )( )( )( )

( ) ( )( )

( )( )

( )( )( )( )

( )( )( )

( )

2

22

22 2

21

11 12

1 1

1 1

1

1

2

,

1

2 22 2

1

1

2

22 2

2

pq

p q n

p

qk

kj k

j

k

L

q kq

qq qj qqk kj k n j k nj k

k j jL

qjj k

jL

T f

fC

fC

αµ

αδ α δ α

α

χ

η

χ

η

χ

η

⋅⋅

+ ′ ′+ +

⋅ ⋅

⋅−

Ω∞=−∞

=−∞

+⋅

+∞ − −− − − −

=−∞ =−∞ =−∞

⋅∞

=−∞

≤ ×

≤

∑∑

∑ ∑ ∑

∑

( ) ( )( )

( )( )( )

*

11

1

2

22

q n

q

qj k n

k jC

δ α +

⋅ ⋅

∞ − −

= +

≤

∑

A. Abdalmonem et al.

1175

where ( )( )

22

*1

mink

q kq

q∈+

=

, this implies that

( )( )( ), 1111 1

q npK

C C f αη η ⋅⋅

≤ ≤

Finally, we estimate 13η by Lemma 3.7, we get

( )( )

( )( )

( )

( )

( )

( ) ( )( )

( )

32 2

22

2

32

2

, ,2 2

1 1

2 1

2 2

,12 d

ppq

jp

q q k

k kj k j k

j k j k

k k

LL

q k

kj kn

k j k CL

T f T fC

x x yf y y

x y

α αµ µ

αµ

χ χ

η η

χη

⋅⋅

⋅

⋅

⋅∞ ∞

Ω Ω∞ ∞= + = +

=−∞ =−∞

∞ ∞

−=−∞ = +

≤

Ω − ≤ −

∑ ∑∑ ∑

∑ ∑ ∫

(1.4)

Note that, when kx C∈ , 2j k≥ + , then ~x y x− . Therefore, applying the generalized Hölder’s In- equality, we have

( ) ( ) ( )( )

( )1

1

, ,d p

pj

j j jn nLC L

x x y x x yf y y f

x y x yµ µχ⋅

′ ⋅− −

Ω − Ω −≤

− −∫

Define the variable exponent ( ) ( )1 1

1 1 1p r p

= +′ ′⋅ ⋅

by Lemma 3.3, then we have

( ) ( ) ( ) ( )( )

1

1

,d , rp

pj

jj jn nLL

C L

x x yf y y f x x y

x y x yµ µχ

⋅

′ ⋅− −

Ω −≤ Ω −

− −∫

( ) ( )

( )( ) ( ) ( ) ( )( )

( )( ) ( ) ( ) ( )

( )( ) ( )

11 12

11 1

1 1

12

1

2

,d

2 d , d

2 2

2 2

j

j

p p nj

n r np p

p p

jnC

rrj n n

j j sL L

jnj n r

j j L L SL L

jnj n r

j jL L

x x yf y y

x y

C f r r x y y

C f

C f

µ

µ

µ

µ

χ σ

χ

χ

′⋅ ⋅ −−

∞ −′⋅ ⋅

′⋅ ⋅

−

− − −

− −×

− −

Ω −

−

′ ′≤ Ω

≤ Ω

≤

∫

∫ ∫

According Lemma 3.4 and the formula ( ) ( )11

1 1 1p rp

= −′′ ⋅⋅

, we have ( ) ( )1 1

1

p pj jr

B B jL LBχ χ

′ ′⋅ ⋅

−

≈ Then we get

( ) ( ) ( ) ( ) ( )1 1,

d 2 p pjj

j nj j Bn L LC

x x yf y y f

x yµ

µ χ⋅ ′ ⋅− −

−

Ω −≤

−∫ (1.5)

From Equations (1.4), (1.5) and using Lemma 3.7, and ( )

( )( )

1

11

1

21

pq

qjj

L

fα χη ⋅

⋅

⋅ ≤

we can obtain

A. Abdalmonem et al.

1176

( )( )

( )( )

( )

( )( )( ) ( )

( )

2

22

32

211

,2

1

2 1

2

2 2

pq

ppj kp n

q

kj k

j k

k

L

q k

jj nkB B LLk j k L

T f

fC

αµ

µα

χ

η

χ χη

⋅⋅

⋅′ ⋅⋅

⋅∞

Ω∞= +

=−∞

∞ ∞− −

=−∞ = +

≤

∑∑

∑ ∑

Note that

( ) ( )2 12 p pk kk

B BL LC µχ χ⋅ ⋅

−≤ see [9].

Then we have

( )( )

( )( )

( )( )( ) ( )

( )

( )

( )( )

( )

( )

( )

( )

2

22

32

111

32

1

11

,2

1

2 1

2 1

2

2

2 2 2 2

2 2

2 2

pq

ppj kp n

pk

p npj

q

kj k

j k

k

L

q k

jk j jn kB B LLk j k L

q k

Bjj kk L

k j k L B L

k j nk

k j k

T f

fC

fC

C

αµ

α µ µ

µα

α

χ

η

χ χη

χ

η χ

⋅⋅

⋅′ ⋅⋅

⋅

⋅⋅

⋅∞

Ω∞= +

=−∞

∞ ∞− −

=−∞ = +

∞ ∞−

=−∞ = +

∞ ∞−

=−∞ = +

≤

≤

≤

∑∑

∑ ∑

∑ ∑

∑ ∑

( )

( )( )

( )

( )( )( )

( ) ( )( )

( )( )

32

2

1

32

1 1

2

1 1

1

1

2 1

22

p n

p q n

q k

j

L

q k

q qjj kk j n

k j kL

f

fC

δ µ

αδ µ α

η

χ

η

⋅

+

⋅ ⋅

−

⋅∞ ∞

− − +

=−∞ = +

≤

∑ ∑

where

( )

( )( )

( )

( )( )

( )( )

( )

( )( )

2

22

2

22

,2

21

32

,2

21

2, 1

2, 1

pq

pq

q

kj k

j k

L

q

kj k

j k

L

T fq

q k

T fq

αµ

αµ

χ

η

χ

η

⋅⋅

⋅⋅

⋅∞

Ω= +

−

⋅∞

Ω= +

+

≤

= >

∑

∑

Since ( ) ( )2 1q q− +≥ and 2nα µ δ> − , as the same 11η we have

A. Abdalmonem et al.

1177

( )( )( ), 1113 1

q npK

C C f αη η ⋅⋅

≤ ≤

This completes the proof Theorem 1. Proof of Theorem 2 Let ( )nb Lipβ∈ , ( )( ) ( )11,q npf Kα ⋅⋅∈ . We write

( ) ( ) ( )k jj j

f x f x f xχ∞ ∞

=−∞ =−∞

= =∑ ∑

From definition of ( )( ) ( ),q npKα ⋅⋅

( )( )( )( )

( )( )

( )( )

2

, 2222

,,

2 ,, inf 0 : 1q n

ppq

qk m

kmk K k

L

b T fb T f α

αµ

µ

χχ η

η⋅ ⋅⋅⋅

⋅∞ Ω

Ω=−∞

= > ≤

∑

Since

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )

2

22

2 2

2 22 2

2

2

,

2

, ,

21 22 23 21

1

,1

22

2 ,

2 , 2 ,

2 ,

pq

p pq q

pq

qk m

k

L

q qk

k m k mk k

j j

L L

qk

k mk

j k

L

b T f

b T f b T f

b T f

αµ

α αµ µ

αµ

χ

η

χ χ

η η η η

χ

η

⋅⋅

⋅ ⋅⋅ ⋅

⋅

⋅

Ω

⋅ ⋅∞ −

Ω Ω=−∞ =−∞

⋅+

Ω= −

≤ ≤ + +

+

∑ ∑

∑

( )

( )( )

( )( )

2

22 2

,2

23

2 ,

pq

q

k mk

j k

L

b T fα µ χ

η⋅

⋅ ⋅

⋅∞

Ω= +

+

∑

where

( )( ) ( )( )2 2

2

21 ,2 ,q p

kk m

kj k l L

b T fα µη χ⋅ ⋅

∞−

Ω=−∞ =−∞

= ∑

( )( ) ( )( )2 2

1

22 ,1

2 ,q p

kk m

kj k k l L

b T fα µη χ⋅ ⋅

∞+

Ω= − =−∞

= ∑

( )( ) ( )( )2 2

23 ,2

2 ,q p

k mk

j k k l L

b T fα µη χ⋅ ⋅

∞∞

Ω= + =−∞

= ∑

And 21 22 23η η η η= + + . The similar to prove of Theorem 1

( )( )( )( ) [ ], 22, 21 22 23

, q np

mk K

b T f C Cαµ χ η η η η⋅⋅

Ω ≤ ≤ + +

Hence ( ) ( )( )( ), 1121 22 23, , qn n

p

mLip KC b f αβη η η ⋅⋅

≤

. Denote ( )( )( ), 111

q npK

f αη ⋅⋅

=

First we estimate 22η . Note that ,,mb T µΩ is bonuded on

( ) ( )p nL ⋅ (Proposition 3.3), similarly to esti-

A. Abdalmonem et al.

1178

mate for 12η in the proof of the Theorem 1, we get that

( )( )

( )( )

2

22

1

,1

1

2 ,

pq

qk

k mj k

j k

k

L

b T fC

αµ χ

η⋅⋅

⋅+

Ω∞= −

=−∞

≤

∑∑

That is

( ) ( ) ( )( )( ), 1122 1qn n n

p

m mLip Lip KC b C b f αβ βη η ⋅⋅

≤ ≤

Now, we estimate of 21η . Using size condition of jf and Minkowski inequality, when 1j k≤ − we get,

( ) ( ) ( )( ) ( )

( ) ( )

( ) ( )

( )( ) ( )

,,

, d

,d

,d

,d

n

n

n

n

mmn

m mn Lip

mn n mLip

mn n mLip

x x yb T b x b y f y y

x y

x x yx y b f y y

x y

x x yb f y y

x y

x x yb f y y

x y

β

β

β

µ µ

βµ

µ µ β

µ µ β

Ω −

−

− − −

− − +

Ω − = − −

Ω −≤ −

−

Ω −≤

−

Ω −≤

−

∫

∫

∫

∫

We have that

( ) ( )( )

( )( )( )

( )( )2

2

,, ,

, dpj

p n

mmk j kn m n mLip BL

L

x x y x xb T C b f y y

x y xβµ µ β µ βχ χ⋅

⋅

Ω − + − +

Ω − Ω ≤ − −

∫

(1.6)

The similar way to estimate of ,T µΩ in the proof of Theorem 1, we get that

( ) ( ) ( )( )12,, 2 d p x np kjmm kn

k j BLip B LLb T C b f y y

βµχ χ

⋅−

Ω ≤ ∫

(1.7)

By (1.7) and lemma 3.7, we obtain that

( )( )

( )( )

( )( )

( )

( )( )

( )

2 2

22 2

22

11

2 2

, ,

1 1

2 2

1 1

2 , 2 ,

2 2 2

pq p

pn

q qk k

k m k mj k j k

j jm m

k kLip Lip

L L

q kk k

j jk kn kk L

k j k jL L

b T f b T fC

b b

f fC C

β β

α αµ µ

α α

χ χ

η η

χη η

⋅⋅ ⋅

⋅

⋅ ⋅− −

Ω Ω∞ ∞=−∞ =−∞

=−∞ =−∞

∞ − ∞ −−

=−∞ =−∞ =−∞ =−∞

≤

≤ ≤

∑ ∑∑ ∑

∑ ∑ ∑ ∑

( )( )( ) ( )

( )

( )( )( ) ( )

( )

( )( )

( )

( )

( )

( )

( )( )

22

111

2222

1

111 1 1

2

1

1

2 21

1 1

2

1

2

2 2

2 2

ppj kp n

pj

ppj kp pn n pk

p n

q k

knB B LL

q kq kk k Bj jk k L

B B LLk j k j BL L L

qk

jj k nk

k j L

f fC C

fC

α α

δα

χ χ

χχ χ

η η χ

η

⋅′ ⋅⋅

′ ⋅

′ ⋅′ ⋅⋅ ⋅ ′ ⋅

⋅

−

∞ − ∞ −−

=−∞ =−∞ =−∞ =−∞

∞ −−

=−∞ =−∞

≤ ≤

≤

∑ ∑ ∑ ∑

∑ ∑

( )( )( )

( )

( ) ( )( )

( )( )22

21 1

1

1 1

1

2

1

22

p q n

q kk q qjk j kj k n

k jL

fC

αδ α χ

η

+

⋅ ⋅

⋅∞ −

− −

=−∞ =−∞

≤

∑ ∑

A. Abdalmonem et al.

1179

where

( )

( )( )

( )

( )( )

( )( )

( )

( )( )

2

22

2

22

2

,

21

22

,1

21

2 ,, 1

2 ,, 1

pq

pq

qk

k mj k

jmLip

L

q

k mj k

j kmLip

L

b T fq

b

q k

b T fq

b

β

β

αµ

αµ

χ

η

χ

η

⋅⋅

⋅⋅

⋅−

Ω=∞

−

⋅∞

Ω= −

+

≤

=

>

∑

∑

Since ( ) ( )2 1q q− +≥ and 1nα δ> − , the similar way to estimate 11η in the proof of Theorem1, we can obtain that

( )( )

( )( )

( )( )( )

( ) ( )( )

2

22

*

1

1

1 1

2

,

1

2

1

2 ,

22

pq

p q n

qk

k mj k

j

k

Lq

qjk j kj k n

k jL

b T f

fC C

αµ

αδ α

χ

η

χ

η

⋅⋅

⋅ ⋅

⋅−

Ω∞=−∞

=−∞

⋅∞ −

− −

=−∞ =−∞

≤ ≤

∑∑

∑ ∑

where ( )( )

22

*1

mink

q kq

q∈+

=

, this implies that

( ) ( ) ( )( )( ), 1121 1qn n n

p

m mLip Lip KC b C b f αβ βη η ⋅⋅

≤ ≤

Finally, we estimate 23η . Note that, when kx C∈ , 2j k≥ + , then ~x y y− , we can obtain that

( ) ( ) ( )( ) ( )

( ) ( )

( )( ) ( )

,,

, d

,d

,d

n

n

n

mmn

m mn Lip

mn mLip

x x yb T b x b y f y y

x y

x x yx y b f y y

x y

x x yb f y y

x y

β

β

µ µ

βµ

µ β

Ω −

−

− +

Ω − = − −

Ω −≤ −

−

Ω −≤

−

∫

∫

∫

Then we have

( )( ) ( ),

,, dn

mmn mLip

x x yb T b f y y

x yβµ µ βΩ − +

Ω − ≤ −

∫

(1.8)

Applying the generalized Hölder’s Inequality, we get

( )( ) ( ) ( )

( )( )

( )1

1

, ,d p

pj

j j jn m n mLC L

x x y x x yf y y f

x y x yµ β µ βχ⋅

′ ⋅− + − +

Ω − Ω −≤

− −∫

A. Abdalmonem et al.

1180

Define the variable exponent ( ) ( )1 1

1 1 1p r p

= +′ ′⋅ ⋅

by Lemma 3.3, then we have

( )( ) ( )

( ) ( ) ( )( )

( )( )( ) ( ) ( ) ( )( )

( )( )( ) ( ) ( ) ( )

( )( )( ) ( )

1

1

11 12

11 1

1 1

12

1

2

,d

,

2 d , d

2 2

2 2

j

rp

p

j

p p nj

n r np p

p p

jn mC

jj n mLL

L

rrj n m n

j j sL L

jnj n m r

j j L L SL L

jnj n m r

j jL L

x x yf y y

x y

f x x yx y

C f r r x y y

C f

C f

µ β

µ β

µ β

µ β

µ β

χ

χ σ

χ

χ

⋅

′ ⋅

′⋅ ⋅ −−

∞ −′⋅ ⋅

′⋅ ⋅

− +

− +

− − + −

− − +

×

− − +

Ω −

−

≤ Ω −−

′ ′≤ Ω

≤ Ω

≤

∫

∫ ∫

According Lemma 3.4 and the formula ( ) ( )11

1 1 1p rp

= −′′ ⋅⋅

, we have ( ) ( )1 1

1

p pj jr

B B jL LBχ χ

′ ′⋅ ⋅

−

≈ . Then we get

( )( ) ( )

( )( )( ) ( )

( )( )( ) ( )

1 1

1 1

,d 2 2 2

2

p pjj

p pj

jn jnj n m r r

j j Bn m L LC

j n mj BL L

x x yf y y f

x y

f

µ βµ β

µ β

χ

χ

⋅ ′ ⋅

⋅ ′ ⋅

−− − +

− +

− − +

Ω −≤

−

≤

∫

By (1.8), we can obtain that ( )( )

( ) ( )1 1,, 2 p pj

mj n mmj BLip L L

b T C b fβ

µ βµ χ⋅ ′ ⋅

− − +Ω ≤ (1.9)

Then by (1.9) and Lemma 3.7, we have

( )( )

( )( )

( )( )

( )

2 2

22 2

, ,2 2

1 1

2 , 2 ,

pq p

q q

k m k mj k j k

j k j km m

k kLip Lip

L L

b T f b T fC

b bβ β

α αµ µχ χ

η η⋅⋅ ⋅

⋅ ⋅∞ ∞

Ω Ω∞ ∞= + = +

=−∞ =−∞

≤

∑ ∑∑ ∑

where

( )( )

( )

( )( )

( )( )

( )

( )( )

2

22

2

22

,2

21

32

,2

21

2 ,, 1

( )

2 ,, 1

pq

pq

q

k mj k

j kmLip

L

q

k mj k

j kmLip

L

b T fq

b

q k

b T fq

b

β

β

αµ

αµ

χ

η

χ

η

⋅⋅

⋅⋅

⋅∞

Ω= +

−

⋅∞

Ω= +

+

≤

=

>

∑

∑

Furthermore, when ( ) 2m nα µ β δ> + − , note that ( ) ( ) ( )2 12p pk kk m

B BL LC µ βχ χ⋅ ⋅

− +≤ see [9], the similar way to estimate 11η , we get

A. Abdalmonem et al.

1181

( )( )

( )( )

( )( )

( )( )( ) ( )

( )

( )( )

( )( )( )

( )( )

( )

2

22

32

211

32

111

,2

1

2 1

2 1

2 ,

2 2

2 2 2

pq

ppj kp n

ppj kp n

q

k mj k

j km

k Lip

L

q k

j n m jkB B LLk j k L

q k

j n m j k mkB B LLk j k L

b T f

b

fC

fC

C

β

αµ

µ βα

µ β µ βα

χ

η

χ χη

χ χη

⋅⋅

⋅′ ⋅⋅

⋅′ ⋅⋅

⋅∞

Ω∞= +

=−∞

∞ ∞− − +

=−∞ = +

∞ ∞− − + − +

=−∞ = +

≤

≤

≤

∑∑

∑ ∑

∑ ∑

( )( )

( )( )( ) ( )

( )

( )( )

( )( )

( )

( )

( )

( ) ( )( )

( )( )

( )

( )

32

111

32

1

11

32

2

1

2

1

2 1

2 1

2 1

2

2 2

2 2

2 2

2

ppj kp n

pk

p npj

p n

q k

jj k mkB B LLk j k L

q k

Bjj k mk L

k j k L B L

q k

k j n m jk

k j k L

k j n

k j k

f

fC

fC

C

µ βα

µ βα

δ µ βα

δ

χ χη

χ

η χ

η

⋅⋅⋅

⋅

⋅⋅

⋅

∞ ∞ −− +

=−∞ = +

∞ ∞− +

=−∞ = +

∞ ∞− − +

=−∞ = +

∞ ∞−

=−∞ = +

≤

≤

≤

∑ ∑

∑ ∑

∑ ∑

∑ ∑

( )( )( )

( ) ( )( )

( )( )32

1 1

1 1

1

1

2

p q n

q k

q qjj km

L

fαµ β α χη

+

⋅ ⋅

⋅

− + +

We can conclude that

( )( )

( )( )

( ) ( )( )( )

( ) ( )( )

2

22

*

1

2

1 1

,2

1

2 1

2 ,

22

pq

p q n

q

k mj k

j km

k Lip

Lq

qjj kk j n m

k j kL

b T f

b

fC C

β

αµ

αδ µ β α

χ

η

χ

η

⋅⋅

⋅ ⋅

⋅∞

Ω∞= +

=−∞

⋅∞ ∞

− − + +

=−∞ = +

≤ ≤

∑∑

∑ ∑

where ( )( )

32

*1

mink

q kq

q∈+

=

, this implies that

( ) ( ) ( )( )( ), 1123 1qn n n

p

m mLip Lip KC b C b f αβ βη η ⋅⋅

≤ ≤

This completes the proof Theorem 2.

Competing Interests The authors declare that they have no competing interests.

A. Abdalmonem et al.

1182

Acknowledgements This paper is supported by National Natural Foundation of China (Grant No. 11561062).

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http://dx.doi.org/10.1090/cbms/083http://dx.doi.org/10.2307/1992955http://dx.doi.org/10.1080/00036817808839193http://dx.doi.org/10.1215/S0012-7094-86-05313-5http://dx.doi.org/10.1090/S0002-9947-1971-0285938-7http://dx.doi.org/10.1007/s12215-010-0015-1http://dx.doi.org/10.1186/1029-242x-2014-227http://dx.doi.org/10.3906/mat-1412-52http://dx.doi.org/10.2748/tmj/1178224663

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Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz SpacesAbstractKeywords1. Introduction2. Definition of Function Spaces with Variable Exponent3. Properties of Variable Exponent4. Main Theorems and Their ProofCompeting InterestsAcknowledgementsReferences