The Boundedness of Fractional Integral with Variable Kernel on Variable Exponent Herz Morrey Spaces

(1)

http://dx.doi.org/10.4236/jamp.2016.44089

How to cite this paper: Abdalmonem, A., Abdalrhman, O. and Tao, S.P. (2016) The Boundedness of Fractional Integral with Variable Kernel on Variable Exponent Herz-Morrey Spaces. Journal of Applied Mathematics and Physics, 4, 787-795.

http://dx.doi.org/10.4236/jamp.2016.44089

The Boundedness of Fractional Integral with

Variable Kernel on Variable Exponent

Herz-Morrey 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}

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 with variable kernel. Under some assumptions, we prove that such kind of operators is bounded from the variable exponent Herz-Morrey spaces to the variable exponent Herz-Morrey spaces.

Keywords

Fractional Integral, Variable Kernel, Variable Exponent, Herz-Morrey 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 , n

x z∈ , one has Ω

(

x,λz

)

= 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 TΩ_,_µ is defined by

( )

(

) ( )

,

d ,

n n

x x y

T f x f y y

x y

µ µ

Ω −

=

−

∫



(2)

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, TΩ_,_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

, d .

sup _n

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

d . sup _n

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-Litelewood maximal operator.

Many classical results about the fractional integral operator with variable kernel have been achieved [1]-[4]. In 1971, Muckenhoupt and Wheeden [5] had proved the operator TΩ_,_µ was bounded from Lp to Lq. In 1991,

Kováčik and Rákosník [6] introduced variable exponent Lebesgue and Sobolev spaces as a new method for dealing with nonlinear Dirichet boundary value problem. Then, variable problem and differential equation with variable exponent are intensively developed. In last years, more and more researchers have been interested in the theory of the variable exponent function space and its applications. The class of Herz-Morrey spaces with variable exponent is initially defined by the author [7], and the boundedness of vector-valued sub-linear operator and fractional integral on Herz-Morrey spaces with variable exponent was introduced by authors [7] and [8]. We also note that Herz-Morrey spaces with variable exponent are generalization of Morrey-Herz spaces [9] and Herz spaces with variable exponent [10]. Recently, Wang Zijian and Zhu Yueping [11] proved the boundedness of multilinear fractional integral operators on Herz-Morrey spaces with variable exponent.

The main purpose of this paper is to establish the boundedness of the fractional integral with variable kernel from _{( )}

( )

1 1

, ,

n q p

MKα λ _⋅  to _{( )}

( )

2 2

, ,

n q p

MKα λ _⋅  . Throughout this paper E denotes the Lebesgue measure, χ_E means the 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

(3)

Let E be a measurable set in n with E >0. We first defined Lebesgue spaces with variable exponent.

Definition 2.1. Let p

( )

⋅ :E→

[ )

1,∞ be a measurable function. The Lebesgue space with variable exponent ( )

_{( )}

p

L ⋅ 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 . Let Bk =

{

x∈n: x≤2k

}

,Ck =Bk \Bk−₁,

χ

k =

χ

_Bk,k∈

Definition 2.2. Let α∈, 0< < ∞q ,p

( )

⋅ ∈

( )

n and 0≤ < ∞λ . The Herz- Morrey spaces with variable exponent MK_{q p}α λ_,,_{( )}_⋅

( )

n is defined by

( )

{

( )

(

{ }

)

,_{( )}

( )

}

,

, \ 0 : n

q p p

n n

Loc

q p MK

MKα λ f L f α λ ⋅

⋅

⋅ = ∈  < ∞





 

( )

, ,

1

: sup 2 2 p

n q p

q L

q

L k q

k

MK L

L K

f α λ λ α fχ ⋅

⋅

−

∈ =−∞

 

= _ _



∑



 _



Remark 2.1. (See [6]) Comparing the Homogeneous Herz-Morrey Spaces with variable exponent with the homogeneous Herz spaces with variable exponent, where K_qα_{( )},_⋅p

( )

n is defined by

( )

(

{ }

)

_{( )},

( )

_{( )}

1 ,

\ 0 : p n 2 q n

q

p p

q

p n n kp

Loc k

q K L

k

Kα f L f α α fχ ⋅

⋅

∞ ⋅

⋅

=−∞

 _ _ 

 

=_ ∈ =_ _ < ∞_

 

 

 

∑





 

Obviously, Kα_q_{( )},_⋅p

( )

n =MK_{p q}α_,,0_{( )}_⋅

( )

n

3. Properties of Variable Exponent

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

( )

n _and

( ) ( )

n r n1

L∞ L S −

Ω ∈ _ × .

Proposition 3.1. (See [12]) 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

− ≤ ≥

+

(4)

Proposition 3.2. (see [13]) Suppose that p₁

( )

⋅ ∈B

( )

n , Ω ∈L∞

( ) ( )

n ×L Sr 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 p1( )

( )

n

f ∈L ⋅  ,

( )

_{( )}

1( )

_{( )}

2

, _Lp n _Lp n

TΩµf ⋅ _ = f ⋅ _

Now, we need recall some lemmas

Lemma 3.1. (See [14]) Given p

( )

⋅ :n→

[

1,∞

)

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 [15]) Suppose that 0< <

µ

n, r>1, Ω ∈L∞

( ) ( )

n ×L Sr n−1 satisfies the Lr-Dini condition. If there exists an 0< <α 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 [16]) 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

( ) ( )

_Lp( )

_{( )}

n

( )

_Lr

_{( )}

n

( )

_Lq( )

_{( )}

n

f x g x ⋅ _ ≤C g x _ f x ⋅ _

Lemma 3.4. (See [17]) Suppose that p

( )

⋅ ∈B

( )

n and 0< p−≤ p+ < ∞. 1) For any cube and 2n

Q ≤ , all the

χ

∈Q, then: p_{( )} 1 ( ) p x

Q L Q

χ ⋅ ≈

2) For any cube and Q ≥1, then _p_{( )} 1p

Q L Q

χ ∞

⋅ ≈ where

( )

lim_x

p_∞ = _→∞p x

Lemma 3.5. (See [18]) If p

( )

⋅ ∈B

( )

n , then there exist constant C>0 such that for all balls B in n and all measurable subset S ⊂R

( )

_{( )}

( )

_{( )}

( )

1 1

1 ,

p n p n

S L S L

B L B L

S S

C C

B B

δ δ

χ χ

′

⋅ ⋅

′

⋅ ⋅

   

≤ _ _ ≤ _ _

   

 

such that δ δ, ₁ is constants satisfying 0<δ δ, ₁<1

Lemma 3.6. (See [14]) 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 ′⋅  ≤

4. Main Theorem and Its Proof

In this section we prove the boundedness of fractional integral with variable kernel on variable exponent Herz- Morrey spaces under some conditions.

Theorem A. Suppose that 0< <µ n, 0< ≤β 1,λ α< <nδ₁+β, 0<q₁≤q₂< ∞. Let

( ) ( )(

1

)

2

n r n

L∞ L S − r p+

(5)

( )

1 1 0 d r β ω δ δ δ + < ∞

∫

And let p₁

( )

⋅ ∈B

( )

n satisfy

( )

1

0 n

p

µ

+

< ≤ and define the variable exponent p₂

( )

x by

( )

1 2

1 1

p x p x n

µ

− = , then we have

( )

, ( )

( )

,

, 1 1

2 2

, n n .

q p

q p MK

MK

T µf α λ C f α λ ⋅ ⋅

Ω  _ ≤  _

For all _{( )}

( )

1 1

,

, .

n q p

f∈MKα λ _⋅ 

Proof If _{( )}

( )

1 1

, ,

n q p

f∈MKα λ ⋅  arbitrarily, we apply inequality

1 2 1 2 1 1 q q q q j j j j a a ∞ ∞ = =   ≤  



∑



∑

q q1, 2,≥0

( )

_{( )}

1 2

1 ₁ ₂ 2

, 2 , 2 2 1 1 1 2 , , ,

sup 2 2

p n n q p p n q q L

q _{L q} _{k q} q

k

MK L

L K

L _q

L q k q

k L

L K

T f T f

T f

α λ λ α

µ µ λ α µ χ χ ⋅ ⋅ ⋅ − Ω Ω ∈ =−∞ − Ω ∈ =−∞   = _ _     ≤ _ _  

∑

   _  

If we denote

( )

_k _j

( )

,

k k

f x f x χ f x

∞ ∞

=−∞ =−∞

=

∑

=

∑

Then we have

( )

_{( )}

1

1 ₁ ₁

, 2 , 2 2 1 1 1 2 1 1 1 2 , , 2 , , 1 1 2

sup 2 2

.

n p n

q p

p n

q L

q _{L q} _{k q}

j k

MK _L L

k j

q

L k

L q k q

j k L

L k j

q L

L q k q

j k L

L k j k

T f T f

T f

T f U U

α λ λ α

µ µ λ α µ λ α µ χ χ χ ⋅ ⋅ ⋅ ⋅ ∞ − Ω Ω ∈ =−∞ =−∞ − − Ω ∈ =−∞ =−∞ ∞ − Ω ∈ =−∞ = −   ≤ _ _     ≤ _ _     +     = +

∑

       

Below, we first estimate U₁ using size condition of f_j. Minkowski inequality when j≤ −k 1, we get

( )

_{( )}

( )

( ) 2( ) 2

2

1

, p n , d

k

p p

j k _C j

L

T µ f χ ⋅ T µ f x

⋅ ⋅

Ω _ = _

∫

Ω _

( )

(

) ( )

,

d

n

j n j

x x y

T f f y y

x y µ µ Ω − Ω − = −

∫



Then we have

( )

_{( )}

( ) (

)

( )

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

(6)

(

)

( )

_{( )}

(

)

( )

(

)

( )

2 2 2 , , , , , , p n

p x n r n

p x n k r n k n n L k

n n L

L

B

n n _L

L

x x y x x

x y x

x x y x 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 to Lemma 3.4 and the formula

( )

2 2

1 1 1

p x = p x −r

 , then we have

( )

_{( )}

( )

_{( )}

2 1

1

p x

p x n n

k k

r n

B B

L L B

µ

χ χ −−

≈

 _ _ . Combining Lemma 3.2, note that 2 2( )

j k

j k− ≤ − β_{, we get}

(

)

( )

( ) ₍ ₎ 1 2 1 ₂ 1 1 0 1 1 0 1 , , d 2

2 2 d

2 1 d

2 2 k k r n n n _y r r

n n k _y

L

n n

j k r

n n

j k r

r n k n j k r y

x x y x x

CR

x y x

CR CR C µ µ µ µ _β β µ _β β µ _β ω δ δ δ 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

j k _B j B

L L

T _µ f χ ⋅ C β f y y χ

− + −

Ω ≤

∫

_



Using Lemma 3.1, Lemma 3.5 and Lemma 3.6, we obtain

( )

_{( )}

( ) ( )

_{( )}

( )

_{( )}

( )

( ) ( )

_{( )}

( )

_{( )}

( )

_{( )}

( ) ( )

_{( )}

( )

( ) ( ) ( )

_{( )}

( )( ) ( )

_{( )}

1 1 2 1

1 1 1

1 1 1 1 1 1 1 , 2 2 2 2 2 2 2

p x n p x n

p n j p n k

p x n j p n k p x n

p n

j

p n

k

p x n

p n

kn j k

j k j _L B B _L

L L

j k kn

j _L B B _L

L B L j k j L B _L

j k j k n j L

j k n j L

T f C f

C f C f C f f β µ β β β δ β δ

χ χ χ

χ χ χ χ ⋅ ′ ⋅ ′ ⋅ ′ ⋅ ⋅ ′ ⋅ ⋅ − + − Ω − − − − − − + ≤ ≤ ≤ ≤ ≤            

Hence we have

( )( ) ( )

_{( )}

( )( ) ( )

_{( )}

1 1 1 1 1 1 1 1 2 1 2

sup 2 2 2

p x n

p x n q

L k

j k n

L q k

j L

L k j

q

L k

j k n

L q j

j L

L k j

U C f

C f

β δ

λ α

β δ α

(7)

Remark that α <nδ₁+β. We consider the two cases 1<q₁< ∞ and 0<q₁≤1. If 0<q₁≤1, then we use the Hölder inequality and obtain

( )( ) ( )

_{( )}

( )( ) ( )( ) ( )

_{( )}

( )

_{( )}

1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 2 2 1 2 2 2

sup 2 2 2 2

sup 2 2 2

sup 2 2

q q p n q p n p n q q

L k _q k

j k n j k n

L q jq

j L

L k j j

L k _q

j k n

L q jq

j L

L k j

l _q L

L q jq

j L

L j k j

U C f

C f

β δ α β δ α

λ α

β δ α

λ α λ α ′ ⋅ ⋅ ⋅ ′ − − − + − − + − − ∈ =−∞ =−∞ =−∞ − − + − − ∈ =−∞ =−∞ − − ∈ =−∞ = +     ≤   ×      ≤ ≤

∑ ∑

∑

∑ ∑

∑

      ( )( ) ( )

_{( )}

( )

2 1 1 1 1 1 1 1 , , 1 1 2 2

sup 2 2

q

p n

n q p

j k n

l _q

L q jq

j L

L j

q Mk

C f

C f α λ

β δ α

λ α ⋅ ⋅ − + − − − ∈ =−∞ ≤ ≤

∑

_   _

If 0<q₁≤1, α<nδ₁+β, we get

( )( ) ( )

( )

( )( ) ( )

_{( )}

( )

1 1 1 1 1 1 1 1 ₁ 1 1 1 1 1 1 1 1 , , 1 1 2 1 2 2 2

sup 2 2 2

sup 2 2

q p n q p n p n n q p

L k _q

j k n

L q jq

j L

L k j

l _q L

j k n

L q jq

j L

L j k j

l _q

L q jq

j L

L j

q MK

U C f

C f

C f α λ

β δ α

λ α

β δ α

λ α λ α ⋅ ⋅ ⋅ ⋅ − − + − − ∈ =−∞ =−∞ − − + − − ∈ =−∞ = + − − ∈ =−∞ ≤ ≤ ≤ ≤

∑ ∑

∑

       _

Next we estimate U₂, by using Proposition 3.2 we have

( )

_{( )}

( )

_{( )}

( ) ( )

_{( )}

( ) ( )

_{( )}

1 1 1 2 1 1 1 1 1 1 1 1 1 2 , 1 1 1 1 1

sup 2 2

sup 2 2 2

p n p n p n p n q L

L q k q

j k L

L k j k

q L

L q k q

j L

L k j k

q L

k j

L q j

j L

L k j k

L k

k j

L q j

j L

L k j k

U C T f

C f C f C f λ α µ λ α α λ α α λ α χ _⋅ ⋅ ⋅ ⋅ ∞ − Ω ∈ =−∞ = − ∞ − ∈ =−∞ = − ∞ − − ∈ =−∞ = − + − − ∈ =−∞ = −   ≤ _ _     ≤ _ _     ≤ _ _   ≤

∑

∑ ∑

        ( ) ( )

( )

1 1 1 1 2 1 2

sup 2 2 2 p n

q

q L

k j

L q j

j L

L k j k

C f J J α λ α ⋅ ∞ − − ∈ =−∞ = +         +     = +

∑ ∑

_ 

First we estimate of J₁, then we have

( ) ( )

_{( )}

( )

_{( )}

( )

1 1 1 1 1 1 1 1 , , 1 1 1 1 1

sup 2 2 2

sup 2 2

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

L q j

j L

L k j k

L

q

L q kq

k L

L k

q MK

J C f

C f

C f α λ

(8)

To estimate of J₂, when λ α< , we have ( )

( )

_{( )}

( ) ( )( )

( )

_{( )}

( ) ( )( )

( )

_{( )}

1 1

1

1 1

1

1 1 1 1

1

2

1

2

sup 2 2 2

sup 2 2 2 2

sup

p n

q L

k j

L q j

j L

L k j k

q L

K L q k j j j

j L

L k j k

q j

L _q

q

K L q k j j n

n L

L k j k n

L k

J C f

C f

C

α

λ α

λ α λ λ α

⋅

∞

− −

∈ =−∞ = +

∞

− − − −

∈ =−∞ = +

∞

− − − −

∈ =−∞ = + =−∞

∈ =

 

≤ _ _

 

 

≤ _ _

 

 

 

 

≤ _ _ _ _

 

 

 

≤

∑ ∑

∑

 



( ) ( )( )

( )

1

1 1

1

, ,

1 1 1 1

2

2 2 n n

q p q p

q L

q q

K L q k j

MK MK

j k

f α λ C f α λ

λ α λ

⋅ ⋅

∞

− − −

−∞ = +

 

≤

 

 

∑

 _  _

Complete prove Theorem A.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

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

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