Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz Spaces

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 VaVa-riable 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 Abdalmonem

1,2

, Omer Abdalrhman

1,3

, Shuangping Tao

1*

1_{College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China} 2_{Faculty of Science, University of Dalanj, Dalanj, Sudan}

3_{College 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< <

µ

n, _{Ω ∈L}∞

( ) ( )

n _{×L S}r n−1

(

_r≥1

)

_{is homogenous of degree zero on} n , n1

S − denotes the unit sphere in n. If

(i) For any _{x z}, _∈n_{, one has} Ω

(

_x,λ_z

)

= Ω

( )

_{x z, ;}

(ii)

_{( ) ( )}

1

(

1

(

)

( )

)

1

: sup , d

n r n n

n

r _r

L L S s

x

x z σ z

∞ _× − −

∈

′ ′

Ω _ =

_∫

Ω < ∞



The fractional integral operator with variable kernel T_Ω,µ is defined by

( )

(

) ( )

,

,

d ,

n n

x x y

T f x f y y

x y

µ µ

Ω −

Ω −

=

−

∫



The commutators of the fractional integral is defined by

( )

(

) ( ) ( )

(

)

( )

,

,

, n d ,

m m

n

x x y

b T f x b x b y f y y

x y

µ µ

Ω −

Ω −

  = −

 

∫

 ₋

When

µ

≡1, the above integral takes the Cauchy principal value. At this time

µ

≡0, TΩ_,µ is much more

close related to the elliptic partial equations of the second order with variable coefficients. Now we need the further assumption for Ω

( )

x z, . It satisfies

(

) ( )

1 , d 0,

n

n s−Ω x z′ σ z′ = ∀ ∈x

∫



For r≥1, we say Kernel function Ω

( )

x z, satisfies the Lr-Dini condition, if Ω meets the conditions (i), (ii) and

( )

1

0 d

r ω δ

δ δ < ∞

∫

where ω δ_r

( )

denotes the integral modulus of continuity of order r of Ω defined by

( )

(

1

(

)

(

)

( )

)

1

,

sup n , , d

n

r _r

r _s

x

x z x z z

ρ δ

ω δ − ρ σ

∈ <

′ ′ ′

=

_∫

Ω − Ω



where

ρ

is the a rotation in n

1 sup

n

z S

z z

ρ ρ

−

′∈

′ ′

= −

when Ω ≡1, T_Ω,µ is the fraction integral operator

( )

( )

, n n d .

f y

T f x y

x y

µ µ

Ω = −

−

∫



The corresponding fractional maximal operator with variable kernel is defined by

( )

(

)

,

0

1

sup _n , d .

x y r r

M f x x x y y

r

µ µ

Ω = _> −

∫

− < Ω −

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

( )

( )

,

0

1

sup _n d

x y r r

M f x f y y

r

µ µ

Ω − _{− <} >

=

_∫

Especially, in the case

µ

≡0, 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 TΩ_,µ was bounded from

p

L to Lq. 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 TΩ_,µ 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 ,

TΩµ and their commutators , ,

m

b TΩµ

 

Throughout this paper E denotes the Lebesgue measure, χ_E means he characteristic function of a measurable set S⊂n. 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 E >0. We first define the Lebesgue spaces with variable exponent. Definition 2.1. see [1] Let p

( )

⋅ :E→

[

1,∞

)

be a measurable function. The Lebesgue space with variable exponent Lp( )⋅

( )

E is defined by

( )

_{( )}

( )

( )

is measurable : d for some constant 0

p x p

E

f x

L E f x η

η

⋅ ₌ _ _{ < ∞ >} 

 _{ } 

   



∫



The space Lp_loc( )⋅

( )

E is defined by

( )

_{( )}

{

_{is measurable :} ( )

_{( )}

_{for all compact}

}

p p

loc

L ⋅ E = f f ∈L ⋅ K K⊂E

The Lebesgue spaces Lp( )⋅

( )

E is a Banach spaces with the norm defined by

( )_{( )}

( )

( )

inf 0 : d 1

p

p x

L E E

f x

f η x

η ⋅

 _{ } 

 

=  > _ _ ≤ 

   



∫



We denote

( )

{

}

essinf : ,

p₋= p x x∈E p₊ =esssup

{

p x

( )

:x∈E

}

. Then 

( )

E consists of all p

( )

⋅ satisfying p−>1 and p+ < ∞.

Let M be the Hardy-Littlewood maximal operator. We denote B

( )

E to be the set of all function

( )

( )

p ⋅ ∈ E satisfying the M is bounded on Lp( )⋅

( )

E .

Definition 2.2. see [17] Let p

( ) ( )

⋅ ,q ⋅ ∈

( )

E . The mixed Lebesgue sequence space with variable exponent

( )

( )

( )

q p

l ⋅ L ⋅ is the collection of all sequences

{ }

0 j _j

f ∞₌ of the measurable functions on _n _{such that}

{ }

_{0 ( )}

_{( )}

_{( )} ( )

( )

( )

0

inf 0 : _{q p} 1

q p

j j _j

l L l L

j

f

f η Q

µ ⋅ ⋅ ⋅ ⋅

∞ ∞

=

=

 _{ }  

 _{ } 

= _ > _{ } ≤ _< ∞

_{ } 

 _{ } 

 

( )

( )

( )

(

{ }

)

( )

( ) ( )

1 0

0

inf ; n d 1

q p

p x

j

j _j j _R

l L

j _{q x}

j

f x

Q f µ x

µ ⋅ ⋅

∞ ∞

= ₌

 _{ } 

 _{ } 

 

=    ≤ 

 

 _{ } 

   

 

∑

∫

Noticing q+< ∞, we see that

( )

( )

( )

(

{ }

)

( )

( ) ( )

0 0

p q p

q

q

j _j j

l L _L

j

Q f f ⋅

⋅ ⋅ _⋅

∞

∞ ⋅

= =

∑

₌

Let

{

n: 2k

}

, \ ₁, ,

k k k k k _Ck

B = x∈ x ≤ C =B B₋ χ =χ k∈

Definition 2.3. see [16] Let α∈,p

( ) ( )

⋅ ,q ⋅ ∈

( )

n . The homogeneous Herz space with variable ex- ponent _{( )},q( )

( )

n

p

Kα_⋅ ⋅  is defined by

( )( )

( )

{

( )

(

{ }

)

_{( )},( )

_{( )}

}

,

\ 0 : q n

p

q n p n

Loc

p K

Kα f L f α ⋅ ⋅

⋅ ⋅

⋅ = ∈   < ∞

 _{ }

( )( )

( )

{

}

( )

( )

( )

( )

( ) ( )

,

0

2

2 inf 0 : 1

q n

p q p

p

q

q k

k k

k

K _k

l L k

L

f

f α f

α

α _{χ η} χ

η ⋅

⋅ ⋅ ⋅

⋅

⋅

⋅ ∞

∞

= _=−∞

 _{ } 

 

= =  > _ _ ≤ 

   

 

∑

 

Remark 2.1. see [16] (1) If _q1

( ) ( )

⋅ ,_q2 ⋅ ∈_

( )

_n _satisfying

( ) ( )

1 2

q ₊≤ q ₊, then

( ),q1( )

( )

n ( ),q2( )

( )

n

p p

Kα_⋅ ⋅  ⊂Kα_⋅ ⋅ 

(2) If _q1

( ) ( )

_⋅ ,_{q2 ⋅ ∈}

( )

_n _and

( ) ( )

1 2

q ₊≤ q ₊, then

( )

( )

( )

2

1

n

q q

⋅ ∈

⋅   and

( )

( )

2

1 1

q q

⋅ ≥

⋅ . Thus, by Lemma 3.7 and Remark 2.2, for any _{( )},q( )

( )

n

p

f ∈Kα_⋅ ⋅  , we have

( )

( ) ( )

( )

( )

( ) ( )( )

*

2 1 1

1

2 1

( )

2 2 2

1

h h

p

p p

q

q q

p

p p

q q q

k k k

k k k

k k k

L

L L

f f f

α _χ α _χ α _χ

η ⋅ η ⋅ η ⋅

⋅

⋅ ⋅

⋅ ⋅ ⋅

∞ ∞ ∞

=−∞ =−∞ =−∞

 

    _   _

≤ ≤ ≤

       

     

 

    _   _

∑

∑

∑

where

( )

( )

( )

( )

2

1

2

1

2

, 1

2

, 1

k k

h _k

k

f q

q p

f q

q

α

α

χ η

χ η

−

+

 ⋅ 

≤ _ _

⋅

 

=   ⋅ 

 _>

 

 _⋅ 

 



0 *

0 min , 1 max , 1

h h

h

h

h h

h

h

p a p

p a ∞

∈ ₌

∞

∈ ₌

 _≤

 = 

 _>



∑

∑





This implies that _{( )},q1( )

( )

n _{( )},q2( )

( )

n

p p

Kα_⋅ ⋅  ⊂Kα_⋅ ⋅  . Remark 2.2. Let h∈,ah≥0,1≤ph< ∞. then

*

0 0

p

h h

h h

a a

∞ ∞

= =

 

≤ _ _

∑

∑

where

0 *

0 min , 1 max , 1

h h

h _h

h h

h

h

p a p

p a ∞

∈ ₌

∞

∈ ₌

 _≤

 = 

 _>



∑

∑





Definition 2.4. see [18] For 0< ≤

β

1, the Lipschitz space

( )

n

Lipβ  is defined by

( )

( )

( )

, ,

: sup

n

n

Lip

x y x y

f x f y

Lip f f

x y

β

β β

∈ ≠

 − 

 

=_ = < ∞_

−

 

  

 (1.1)

3. Properties of Variable Exponent

In this section we state some properties of variable exponent belonging to the class

( )

n



B and

( ) ( )

n r n1

L∞ L S −

Ω ∈ _ × .

Proposition 3.1. see [1] If

( )

( )

n

p ⋅ ∈  satisfies

( )

( )

₍

₎

, 1 2

log

C

p x p y x y x y

−

− ≤ − ≤

( )

( )

₍

₎

, log

C

p x p y y x e x

− ≤ ≥

+ then, we have

( )

( )

n

p ⋅ ∈B  .

Proposition 3.2. see [15] Suppose that _p1

( )

⋅ ∈_B

( )

_n _, Ω ∈_L∞

( ) ( )

_n ×_{L S}r n−1 _{. Let}

( )

1

0 n

p

µ

+

< ≤ , and

define the variable exponent p₂

( )

⋅ by:

( )

( )

1 2

1 1

=

p x p x n

µ

− . Then we have that for all _f ∈_Lp1( )⋅

( )

n _,

( )

_{( )}

1( )

_{( )}

2

, Lp n Lp n

TΩ_µf ⋅ _ ≤c f ⋅ _

Proposition 3.3. Suppose that ₁

( )

( )

n

p ⋅ ∈B  ,

( )

n

b∈Lipβ  , 0< ≤

β

1, Ω ∈L∞

( ) ( )

n ×L Sr n−1 . Let

( )

1

0 m n

p

µ β

+

< + ≤ , and define the variable exponent p₂

( )

⋅ by:

( )

( )

1 2

1 1 m

p x p x n

µ+ β

− = . Then

( )

_{( )}

1( )

( )

2

,

, p n p n

m m

Lip L L

b T f C b f

β

µ _⋅ ⋅

Ω

  ≤

  _ 

Proof

( )

(

) ( ) ( )

(

)

( )

(

_{) ( ) ( )}

₍

₎

_{( )}

(

_{) ( )}

( )

,

,

,

, d

,

d

,

d

n

n

n

m m

n

m n

n m Lip

m Lip

x x y

b T f x b x b y f y y

x y

x x y

b x b y f y y

x y

x x y

C 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

( )

_{( )}

( )

( ) 1( )

( )

2 2

, ,

, _{p n p} p n .

m m

m

Lip Lip L

L L

b T f C b T f C b f

β β

µ _⋅ µ β _⋅ ⋅

Ω Ω +

  ≤ ≤

  _ 

Now, we need recall some lemmas

Lemma 3.1. see [13] Given

( )

: n

[

1,

)

p ⋅  → ∞ have that for all function f and g,

( ) ( )

d p( )

_{( )}

n p( )

_{( )}

n

n f x g x x≤C f L ⋅ g L′⋅

∫

  

Lemma 3.2. see [19] Suppose that 0< <µ n, r>1, _{Ω ∈L}∞

( ) ( )

_n _{×L S}r n−1 _{satisfies the} r

L -Dini con- dition. If there exists an 0<α₀<1 2 such that y <α₀R then

(

)

( )

( )

1

2

2

, ,

d d

r _r

n _n y R

r r

n n

R x R

y R

y x x y x x

x CR

R x y x

µ

µ µ

ω δ δ δ

 _{− +}     

− −

< <

 _{Ω − Ω}  _{ }

 ₋  _{≤  + }

 

 ₋ 

 



∫



∫

Lemma 3.3. see [20] Suppose that x∈n, 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( )

( )

n

( )

q

( )

n

( )

q( )

( )

n

L L L

f x g x ⋅ _ ≤C g x _ f x ⋅ _

Lemma 3.4. see [21] Suppose that _p

( )

_{⋅ ∈B}

( )

_n

1) For any cube and 2n

Q ≤ , all the

χ

∈Q, then: _{p( )} 1p x( )

Q L Q

χ ⋅ ≈ 2) For any cube and Q ≥1, then p_{( )} 1

p Q L Q

χ ∞

⋅ ≈ where

( )

lim_x

p_∞ = _→∞ p x

Lemma 3.5. see [22] If

( )

( )

n

p ⋅ ∈B  , then there exist constants δ δ₁, ₂,C>0 such that for all balls B in n

 and all measurable subset S⊂R

( )

_{( )}

( )

_{( )}

( )

_{( )}

( )

_{( )}

( )

_{( )}

( )

_{( )}

1 1

1

2 1

1

, ,

p n p n

p

p n n

p n

p n

S

B L L

S L B L

S L

B L

B S

C C

S B

S C

B

δ

δ

χ χ

χ χ

χ χ

′

⋅ ⋅

′ ⋅ ⋅

⋅

⋅

 

≤ ≤ _ _

 

  ≤ _ _  

 

 

 

Lemma 3.6. see [13] If p

( )

⋅ ∈B

( )

n , there exist a constant C>0 such that for any balls B in n. we have

( )

_{( )}

( )

_{( )}

1

p n p n

B L B L C

B χ ⋅  χ ′⋅  ≤

Lemma 3.7. see [16] Let p

( ) ( )

⋅ ,q ⋅ ∈

( )

n . If f∈Lp( ) ( )⋅q⋅ , then

( ) ( ) ( ) ( )

(

)

( ) _{( )}

(

( ) ( ) ( ) ( )

)

min p q , p q max p q , p q

p

q q q q q

L L _L L L

f +⋅ ⋅ f −⋅ ⋅ f _⋅ f +⋅ ⋅ f −⋅ ⋅

⋅

≤ ≤

4. Main Theorems and Their Proof

Theorem 1. Suppose that 0< <

µ

n, µ−nδ₂< <α nδ₁, Ω ∈L∞

( ) ( )(

n ×L Sr n−1 r>p₂+

)

,

( ) ( )

( )

1 , 2

n

q ⋅ q ⋅ ∈  with

( ) ( )

q_{2 −}≥ q_{1 +}. And let _p1

( )

_{⋅ ∈B}

( )

_n _satisfy

( )

1

0 n

p

µ

+

< ≤ and define the vari-

able exponent p₂

( )

x by

( )

( )

1 2

1 1

p x p x n

µ

− = . Then the operators TΩ_,µ is bounded from 1( )1( )

( )

,q n

p

Kα _⋅ ⋅  to

( )2( )

( )

2

,q n

p

Kα _⋅ ⋅  .

Theorem 2. Let _{b Lip}

( )

n , ._m

β

∈ _ ∈_ Suppose that 0< <µ n,

(

µ+mβ

)

−nδ₂ < <α nδ₁,

( ) ( )(

1

)

2

n r n

L∞ L S − r p+

Ω ∈  × > , _q1

( ) ( )

_⋅ ,_{q2 ⋅ ∈}

( )

_n _with

( ) ( )

2 1

q ₋≥ q ₊. If _p1

( )

_{⋅ ∈B}

( )

_n _satisfy

( )

1

0 m n

p

µ β

+

< + ≤ and define the variable exponent p₂

( )

x by

( )

( )

1 2

1 1 m

p x p x n

µ+ β

− = . Then the com- mutators bm,TΩ_,µ is bounded from 1( )1( )

( )

,q n

p

Kα_⋅ ⋅  to _{( )}2( )

( )

2

,q n

p

Kα _⋅ ⋅  .

Proof of Theorem1: Let _{( )}1( )

( )

1

,q n p

f ∈Kα_⋅ ⋅  . We write

( )

( )

j j

( )

j j

f x f x χ f x

∞ ∞

=−∞ =−∞

=

∑

=

∑

From definition of _{( )},q( )

( )

n p

Kα_⋅ ⋅ 

( )( )

( )

( )

( )

( ) ( )

2

, 2

2 2

2

, ,

2

, q n inf 0 : 1

p p

q q k

k K

k

L

T f T f α

α µ µ

χ η

η ⋅

⋅ ⋅

⋅

⋅

∞ _Ω

Ω

=−∞

 _{ } 

 _{ } 

=  > _{ } ≤ 

   

 

∑

 _

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

2

2 2

2 2

2 2

2 2

2

2 2

,

2

, ,

11 12 13 11

1

, ,

1 2

12 13

2

2 2

2 2

p q

p p

q q

p q q

k

k

L

q q

k

k k

j k j k

j j

L L

q k

k k

j k j k

j k j k

L

T f

T f T f

T f T f

α µ

α α

µ µ

α α

µ µ

χ η

χ χ

η η η η

χ χ

η η

⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅ Ω

⋅ ⋅

∞ −

Ω Ω

=−∞ =−∞

⋅

+ ∞

Ω Ω

= − = +

 

 

 

 

   

   

   

≤ _{ } ≤ _{ }

+ +

   

   

   

  

 

 

+ _{ } +

 

 

 

∑

∑

∑

∑

( )

( ) ( )

2

2 2

p q q

L ⋅ ⋅

⋅



 

 

 

 

 

 

where

( )

( )

( )

( )

2 2

2

11 2 ,

q p

k k

j k j

k _{l L}

T f

α µ

η χ

⋅ ⋅

∞ −

Ω

=−∞ _=−∞

 

 

=  

 



∑



( )

( )

( )

( )

2 2

1

12 ,

1

2

q p

k k

j k j k

k _{l L}

T f

α

µ

η χ

⋅ ⋅

∞ +

Ω

= − _=−∞

 

 

=  

 



∑



( )

( )

( )

( )

2 2

13 ,

2

2

q p

k

j k j k

k

l L

T f

α

µ

η χ

⋅ ⋅

∞ ∞

Ω

= + _=−∞

 

 

=  

 



∑



And η η= ₁₁+η₁₂+η₁₃, thus

( )

( )

( ) ( )

2

2 2

,

2

p q

q k

j k

k

L

T f

C

α

µ χ

η ⋅

⋅

⋅

∞ _Ω

=−∞

 

  _≤

 

 

∑

That is

( )

( )( )

( )

[

]

, 2 2

, q n 11 12 13 .

p K

T µ f α ⋅ Cη C η η η

⋅

Ω _{ } ≤ = + +

This implies only to prove

( )( )

( )

, 1 1

11, 12, 13 q n

p

K

C f α

η η η

⋅

⋅

≤ _{ } . Denote

( )( )

( )

, 1 1

1 q n

p

K

f α

η

⋅

⋅

= _{ }

Now we consider η₁₂. Applying Lemma 3.7

( )

( )

( ) ( )

( )

( )

( )

( )

( )

( )

1

2 2

2

2 2

1 2

2

1 1

, ,

1 1

1 1

1

,

1 1

2 2

2 p

p q

p

q q k

k k

k k

j k j k

j k j k

k k

L L

q k k

k

j k

k j k

L

T f T f

C

T f C

α α

µ µ

α µ

χ χ

η η

χ η ⋅

⋅ ⋅

⋅

⋅

+ +

Ω Ω

∞ _{= −} ∞ _{= −}

=−∞ =−∞

∞ + _Ω

=−∞ = −

 

 

  _≤

 

 

 

 

 

 

≤ _{ }

 

 

∑

∑

∑

∑

where

( )

( )

( )

( )

( ) ( )

( )

( )

( )

( ) ( )

2

2 2

2

2 2

1

, 1 2

1

1 2

1

, 1 2

1 2

, 1

2

, 1

p q

p q q k

k

j k j k

L q k

k

j k j k

L

T f q

q k

T f q

α

µ

α

µ

χ

η

χ

η

⋅ ⋅

⋅ ⋅

⋅ +

Ω = − −

⋅ +

Ω = − +

 _{ }

 _{ }

 _{ }

≤

 _{ }

 _{ }

 

 _{ }

 = 

  

  

   _>

  

 _ _

 _{ }



∑

∑

By the Proposition 3.2, we get

( )

( )

( ) ( )

( )

( )

2

1 2

1 2

2

1

, ₁

1

1

1 1

2

2

p p

q

q k

k

q k k

j k _k

j k j

k k j k

L

L

T f

f C

α

α µ χ

η η _⋅

⋅ ⋅

⋅ +

Ω

∞ _{= −} ∞ +

=−∞ =−∞ = −

 

 

  _≤

 

 

 

 

∑

∑

∑ ∑

Since _{( )}1( )

( )

1

,q n p

f ∈Kα_⋅ ⋅  , then we have

( )

1

2

1

p

j j

L

f

α _χ

η _⋅ ≤ , and

( )

( ) ( )

1

1 1

1 2

1 p q q j

j j

L

f

α _χ

η ⋅

⋅

⋅ ∞

=−∞

 

  ≤

 

 

∑

By Lemma 3.7 and Remark 2.2, we get

( )

( )

( ) ( )

( )

( ) ( )

( )

( )

( )

( ) ( )

2 1

2

1 1

1 1 2

2

*

1

1

1

1

, 1

1 1

1 2

2

2

p q p

q

p q q

k q k

k

q q

j k k

j k k

k k _L

L L

q q

k k k

L L

T f

f C

f

C C

α

µ α

α

χ

χ

η η

χ η

+

⋅ ⋅ ⋅

⋅

⋅ ⋅

⋅ +

⋅ Ω

∞ _{= −} ∞

=−∞ =−∞

⋅ ∞

=−∞

 

  _{ }

  _{≤  }

  _ _

 

 

 

 

 _{ } 

 

≤ _ _ ≤

 _{ } 

 

 

∑

∑

∑

∑

Hence

( ) ( )

( )

1

1 , 2 2

p ₊ p ₋≤ q k, and

( )

( )

1 2 *

1 min

k

q k q

q ∈

+

=

 , this implies that

( )( )

( )

, 1 1

12 1 q n

p

K

C C f α

η

η

⋅

⋅

≤ ≤ _{ }

Now, we estimate of η₁₁ using size condition of f_j and Minkowski inequality, when j≤ −k 1 we get,

( )

( )

( )

( ) (

)

( )

( )

_{( )}

2

2

,

, ,

d

p n j

p n

j k _{L B} j n n k

L

x x y x x

T f f y y

x y x

µ χ ⋅ µ µ χ

⋅

Ω − −

Ω − Ω

≤ −

−

∫



Since r> p₂+ we define the variable exponent

( )

( )

2 2

1 1 1

p x = +r p x , by Lemma 3.3 we get

(

)

( )

( )

_{( )}

(

)

( )

( )

( )

_{( )}

(

)

( )

( )

( )

( )

2

2

2

, , , ,

, ,

p x n r n

p n

p x n k r n

k k

n n n n L

L L

B

n n _L

L

x x y x x x x y x x

x y x x y x

x x y x x x 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 2

1 1 1

p x = p x −r, then we have

( )

_{( )}

( )

_{( )}

( )

_{( )}

2 2 1

1 1

p x n p x n p x n

k k k

r r n

B _L B _L Bk B _L Bk

µ

χ χ − χ −−

≈ ≈

 _{  } (1.2)

By Lemma 3.2, we get

(

)

( )

( )

( )

( )

( )

( )

1

2

1 2

1

0

1

, ,

d 2

2 1 d

2

k

k r n

n y

n

r r

n n k

y L

n n

j k r

r

n

k n

r

y

x x y x x

CR

x y x

CR

C

µ

µ µ

µ _β

µ

ω δ δ δ

ω δ δ δ −

 _{− +}     

− − −

 _{− +}    ₋  

  − _ − + _

 

Ω − Ω _{ }

− ≤ +

 

− _{ }

 

≤  + 

 

≤

∫

∫



It follows that

( )

2( )

_{( )}

( )

1( )

_{( )}

, p n 2 _j d k p x n

kn

j k _B j B _L

L

T µ f χ ⋅ C f y y χ

−

Ω _ ≤

∫

_ (1.3)

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

( )

( )

( ) ( )

( )

( )

( )

( )

( )

( )

( )

_{( )}

( )

( )

2

2 2

2

2 2

2 2

2 2

1 1 1( )

1 1

2 2

, ,

1 1

2 2

1 1

2 2

2 2 2 2

p

p q

p _{p p}

j k

p

n n

q q k

k k

k k

j k j k

j j

k k

L L

q k q k

k k

j j

k kn k kn

k L B L B L

k j _L k j _L

T f T f

C

f f

C C

α α

µ µ

α α

χ χ

η η

χ χ χ

η η

⋅

⋅ ⋅

⋅ _{′ ⋅ ⋅}

⋅

⋅

− −

Ω Ω

∞ _=−∞ ∞ _=−∞

=−∞ =−∞

∞ − ∞ −

− −

=−∞ =−∞ =−∞ =−∞

 

 

  _≤

 

 

 

 

   

   

≤ ≤

   

   

≤

∑

∑

∑

∑

∑

∑

∑

∑

 

( )

( )

( ) ( )

( )

( )

( )

( )

( )

( )( )

( )

( ) ( )

_{( )}

( )

( )

2 2

2 2

1 ₁

1 ₁ 1

2 2

1 1

1

1 1

2 2

1 1

1

2

1

2 2 2

2 2

p j

p n _p p n

k

p q n

q k q k

k _B k

j j k n j

k L k

k j _{L B} k j _L

L

q k

q q

j k

j k j k n

k j

L

f f

C C

f C

δ

α α

α δ α

χ

η χ η

χ η

′ ⋅

⋅ _{′ ⋅} ⋅

+

⋅ ⋅

∞ − ∞ −

−

=−∞ =−∞ =−∞ =−∞

⋅ ∞ −

− − =−∞ =−∞

   

  _≤  

   

   

 

 

   

   

≤

 

 _{ } 

 

 

∑

∑

∑

∑

∑ ∑

 



( )

( )

( )

( ) ( ) ( )

( )

( )

( ) ( ) ( ) 2 2 2 2 2 2 2 , 2 1 2 2 2 , 2 1 2

, 1

2

, 1

p q p q q k k j k j L q k k j k j L T f q q k T f q α µ α µ χ η χ η ⋅ ⋅ ⋅ ⋅ ⋅ − Ω =−∞ − ⋅ − Ω =−∞ +  _{ }  _{ }  _{ } ≤  _{ }  _{ }    _{ }  =       

   _>

    _ _  _{ } 

∑

∑

Since _{( )}1( )

( )

1

,q n p

f ∈Kα_⋅ ⋅  , then we have

( ) 1 2 1 p j j L f α _χ η ⋅

≤ , and

( ) ( ) ( ) 1 1 1 1 2 1 p q q j j j L f α _χ η ⋅ ⋅ ⋅ ∞ =−∞     ≤    

∑

Now if

( )

q_{1 +} <1, then we have

( )

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

_{( )}

( )

( ) ( ) ( ) ( )

_{( )}

( )( ) 2 2 2 1 1 1 1 1 2 2 * 1 1 1 1 2 , ₂ 1 1 2 1 2 2 2 2 2 p q n p

q

p q n

q k q k q k _q j

j k _k

j k

j j k n

k k j

L L

q q

j

j k j k n

j k j

L T f f C f C C α α µ δ α α δ α χ _χ η η χ η + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ Ω ∞ _=−∞ ∞ − − − =−∞ =−∞ =−∞ ⋅ ∞ ∞ − − =−∞ = +   _{ }   _{   }   _{≤    }   _{   }     _{ }   _{ }           ≤   ≤    _{ }     

∑

∑

∑ ∑

∑

∑

 

where

( )

( )

2 2 * 1 min k q k q q ∈ + = 

If

( )

q_{1 +} ≥1, then we have

( )

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

_{( )}

( )

( ) ( )( )(( ) )

( )

( ) ( ) ( ) 2 2 2 2 2 2 2 1 1 1 1 2 1 1 1 1 1 1 2 , 1 2 2 2 2 1 1 2 2 2 2 2 p q

p q n

p q k k j k j k L q k q q q q j q q

k _{j k n} k _{j k n}

j k

k j j

L q j j k j L T f f C f C α µ α

δ α δ α

α χ η χ η χ η ⋅ ⋅ + _{′ ′} + + ⋅ ⋅ ⋅ − Ω ∞ _=−∞ =−∞ + ⋅ + ∞ − _{− −} − _{− −} =−∞ =−∞ =−∞ ⋅ ∞ =−∞                 _{ }     _{ }   ≤   ×_{ }    _{ }  _{ }           ≤    

∑

∑

∑ ∑

∑

∑

 ( ) ( )

_{( )}

( )( )

( )

* 1 1 1 2 2 2 q n q q j k n

k j

C

δ α +

where

( )

( )

2 2 *

1 min

k

q k q

q ∈

+

=

 , this implies that

( )( )

( )

, 1 1

11 1 q n

p

K

C C f α

η

η

⋅

⋅

≤ ≤ _{ }

Finally, we estimate η₁₃ by Lemma 3.7, we get

( )

( )

( ) ( )

( )

( )

( )

(

_{) ( )}

( )

( )

3

2 2

2

2 2

3 2

2

, ,

2 2

1 1

2 1

2 2

, 1

2 d

p

p q

j

p

q q k

k k

j k j k

j k j k

k k

L L

q k

k

j k

n

k j k C

L

T f T f

C

x x y

f y y x y

α α

µ µ

α

µ

χ χ

η η

χ η

⋅

⋅ ⋅

⋅

⋅

∞ ∞

Ω Ω

∞ _{= +} ∞ _{= +}

=−∞ =−∞

∞ ∞

− =−∞ = +

 

 

  _≤

 

 

 

 

 

Ω −

 

≤ _{ }

−

 

 

∑

∑

∑

∑

∑

∑

∫

(1.4)

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

(

_{) ( )}

( )

(

)

( )

1

1

, ,

d p

p j

j j j

n _L n

C

L

x x y x x y

f y y f

x y µ x y µ

χ ⋅

′ ⋅

− −

Ω − Ω −

≤

− −

∫

Define the variable exponent

( )



_{( )}

1 1

1 1 1

p′ ⋅ = +r p′ ⋅ by Lemma 3.3, then we have

(

_{) ( )}

( )

(

)

( )

1

1 ,

d p , r

p j

j

j j

n L L n

C

L

x x y

f y y f x x y

x y µ x y µ

χ ⋅

′ ⋅

− −

Ω −

≤ Ω −

− −

∫

(

_{) ( )}

( )

( ) ( )

(

(

)

( )

)

( )

( ) _{( )}

( ) ( )

( )

( ) ( )

1

1 1

2

1

1 1

1 1

1 2

1

2 ,

d

2 d , d

2 2

2 2

j

j

p p n

j

n r n

p p

p p

j n C

r r

j n n

j _L j _{L s}

jn

j n _r

j L j L L L S

jn

j n _r

j _L j _L

x x y

f 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 _

( )

1

( )

1

1 1 1

p r

p′ ⋅ = ′ ⋅ − , we have 1( ) 1( )

1

p p

j j

r

B B j

L L B

χ

′⋅

χ

′⋅

−

≈ Then we get

(

_{) ( )}

( )

( ) ( )

1 1

,

d 2 p j p

j

j n

j j B

n L L

C

x x y

f y y f

x y

µ

µ ⋅ χ _{′ ⋅}

− − −

Ω −

≤ −

∫

(1.5)

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

( )

( ) ( )

1

1 1

1 2

1 p q q j

j

L

f

α _χ

η ⋅

⋅

⋅

 

  ≤

 

 

( )

( )

( ) ( )

( )

( )

_{( )}

( ) ( )

( )

2

2 2

3 2

2 1

1

, 2

1

2 1

2

2 2

p q

p p

j k

p n q k

j k j k

k

L

q k j

j n k

B B

L L

k j k _L

T f

f C

α

µ

µ α

χ

η

χ χ

η

⋅ ⋅

⋅ ′ ⋅

⋅

⋅ ∞

Ω ∞ _{= +} =−∞

∞ ∞

− − =−∞ = +

 

 

 

 

 

 

 

 

 

≤

 

 

∑

∑

∑

∑



Note that

( ) ( )

2 2 1

p p

k k

k

B _L C B _L

µ

χ ⋅ ≤ − χ ⋅ see [9]. Then we have

( )

( )

( ) ( )

( )

_{( )}

( ) ( )

( )

( )

( )

_{( )}

( )

( )

( )

( )

2

2 2

3 2

1 1

1

3 2

1

1

1

, 2

1

2 1

2 1

2

2

2 2 2 2

2 2

2 2

p q

p p

j k

p n

p k p n

p j

q k

j k j k

k

L

q k

j

k j jn k

B B

L L

k j k _L

q k

B j

j k

k L

k j k _L

B L

k j n k

k j k

T f

f C

f C

C

α

µ

α µ µ

µ α

α

χ

η

χ χ

η

χ

η _χ

⋅ ⋅

⋅ ′ ⋅

⋅

⋅

⋅

⋅

⋅ ∞

Ω ∞ _{= +} =−∞

∞ ∞

− −

=−∞ = +

∞ ∞

− =−∞ = +

∞ ∞

− =−∞ = +

 

 

 

 

 

 

 

 

 

≤

 

 

 

 

≤

 

 

 

≤

∑

∑

∑

∑

∑

∑

∑

∑





( )

( )

_{( )}

( )

( )( )

( )

( ) ( )

_{( )}

( )

( )

3 2

2

1

3 2

1 1

2

1 1

1

1

2 1

2 2

p n

p q n

q k

j

L

q k

q q

j j k k j n

k j k

L

f

f C

δ µ

α δ µ α

η

χ η

⋅

+

⋅ ⋅

−

⋅ ∞ ∞

− − + =−∞ = +

 

 

 

 

 

   

   

≤

 

 _{ } 

 

 

∑ ∑





where

( )

( )

( )

( )

( ) ( )

( )

( )

( )

( ) ( )

2

2 2

2

2 2

, 2 2

1

3 2

, 2 2

1 2

, 1

2

, 1

p q

p q q k

j k j k

L q k

j k j k

L

T f q

q k

T f q

α

µ

α

µ

χ

η

χ

η

⋅ ⋅

⋅ ⋅

⋅ ∞

Ω = + −

⋅ ∞

Ω = + +

 _{ }

 _{ }

 _{ }

≤

 _{ }

 _{ }

 

 _{ }

 = 

  

  

   _>

  

 _ _

 _{ }



∑

∑

( )( )

( )

, 1 1

13 1 q n

p

K

C C f α

η

η

⋅

⋅

≤ ≤ _{ }

This completes the proof Theorem 1. Proof of Theorem 2

Let _{b Lip}

( )

n

β

∈ _ , _{( )}1( )

( )

1

,q n p

f∈Kα_⋅ ⋅  . We write

( )

( )

k j

( )

j j

f x f x χ f x

∞ ∞

=−∞ =−∞

=

∑

=

∑

From definition of _{( )},q( )

( )

n p

Kα_⋅ ⋅ 

( )

( )( )

( )

( )

( )

( ) ( )

2

, 2

2 2

2

, ,

2 ,

, q n inf 0 : 1

p p

q

q

k m

k m

k K

k

L

b T f

b T f α

α

µ µ

χ

χ η

η ⋅

⋅ ⋅

⋅

⋅

∞ _Ω

Ω

=−∞

 _{   } 

 _   _ 

  = _ > ≤ _

  _{ }

 _{ } 

 

∑

 _

Since

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( )

2

2 2

2 2

2 2

2 2

2

2

,

2

, ,

21 22 23 21

1

, 1

22

2 ,

2 , 2 ,

2 ,

p q

p p

q q

p q q

k m

k

L

q q

k

k m k m

k k

j j

L L

q k

k m

k j k

L

b T f

b T f b T f

b T f

α

µ

α α

µ µ

α

µ

χ η

χ χ

η η η η

χ

η

⋅ ⋅

⋅ ⋅

⋅ ⋅

⋅

⋅ Ω

⋅ ⋅

∞ −

Ω Ω

=−∞ =−∞

⋅ +

Ω = −

 _{ } 

 

 

 

 

   

   

 _{ }   _{ } 

   

≤ _{ } ≤ _{ }

+ +

   

   

   

 

 

 _{ } 

 

+  

 

 

 

∑

∑

∑

( )

( )

( )

( ) ( )

2

2

2 2

, 2

23

2 ,

p q q

k m

k j k

L

b T f

α

µ χ

η

⋅

⋅ ⋅

⋅ ∞

Ω = +

 

 

 _{ } 

 

+  

 

 

 

∑

where

( )

( )

( )

( )

2 2

2

21 2 , ,

q p

k

k m

k j

k _{l L}

b T f

α

µ

η χ

⋅ ⋅

∞ −

Ω

=−∞ _=−∞

 

 _{ } 

=  _{ } 

 



∑



( )

( )

( )

( )

2 2

1

22 ,

1

2 ,

q p

k

k m

k j k

k _{l L}

b T f

α

µ

η χ

⋅ ⋅

∞ +

Ω

= − _=−∞

 

 _{ } 

=  _{ } 

 



∑



( )

( )

( )

( )

2 2

23 ,

2

2 ,

q p

k m

k j k

k _{l L}

b T f

α

µ

η χ

⋅ ⋅

∞ ∞

Ω

= + _=−∞

 

 _{ } 

=  _{ } 

 



∑



And η η= ₂₁+η₂₂+η₂₃. The similar to prove of Theorem 1

( )

( )( )

( )

[

]

, 2 2

, 21 22 23

, _q

n p m

k K

b T _µ f χ _{α ⋅} Cη C η η η

⋅

Ω

  ≤ ≤ + +

   

Hence

_{( )}

( )( )

( )

, 1 1

21, 22, 23 n q n

p m

Lip K

C b f α β

η η η ⋅

⋅

≤ _{  } . Denote

( )( )

( )

, 1 1

1 q n

p

K

f α

η

⋅

⋅

= _{ }

First we estimate η₂₂. Note that bm,TΩ_,µ is bonuded on ( )

( )

p n