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In this paper, we study the boundedness of the fractional integral operator and their commutator on Herz spaecs with two variable exponents . By using the properties of the variable exponents Lebesgue spaces, the boundedness of the fractional integral operator and their commutator generated by Lipschitz function is obtained on those Herz spaces.

Let

(i) For any

The fractional integral operator with variable kernel

The commutators of the fractional integral is defined by

When

For

where

where

when

The corresponding fractional maximal operator with variable kernel is defined by

We can easily find that when

Especially, in the case

Many classical results about the fractional integral operator with variable kernel have been achieved [

Recently, Wang and Tao [

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

Throughout this paper

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

Definition 2.1. see [

exponent

The space

The Lebesgue spaces

We denote

Then

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

Definition 2.2. see [

Noticing

Let

Definition 2.3. see [

where

Remark 2.1. see [

(2) If

and Remark 2.2, for any

where

This implies that

Remark 2.2. Let

where

Definition 2.4. see [

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

Proposition 3.1. see [

then, we have

Proposition 3.2. see [

Proposition 3.3. Suppose that

Proof

By Proposition 3.2, we get

Now, we need recall some lemmas

Lemma 3.1. see [

Lemma 3.2. see [

Lemma 3.3. see [

then for all measurable function f and g, we have

Lemma 3.4. see [

1) For any cube and

2) For any cube and

Lemma 3.5. see [

Lemma 3.6. see [

Lemma 3.7. see [

Theorem 1. Suppose that

able exponent

Theorem 2. Let

mutators

Proof of Theorem1:

Let

From definition of

Since

where

And

That is

This implies only to prove

Now we consider

where

By the Proposition 3.2, we get

Since

By Lemma 3.7 and Remark 2.2, we get

Hence

Now, we estimate of

Since

According Lemma 3.4 and the formula

By Lemma 3.2, we get

It follows that

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

where

Since

Now if

where

If

where

Finally, we estimate

Note that, when

Define the variable exponent

According Lemma 3.4 and the formula

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

Note that

Then we have

where

Since

This completes the proof Theorem 1.

Proof of Theorem 2

Let

From definition of

Since

where

And

Hence

First we estimate

mate for

That is

Now, we estimate of

We have that

The similar way to estimate of

By (1.7) and lemma 3.7, we obtain that

where

Since

where

Finally, we estimate

Then we have

Applying the generalized Hölder’s Inequality, we get

Define the variable exponent

According Lemma 3.4 and the formula

By (1.8), we can obtain that

Then by (1.9) and Lemma 3.7, we have

where

Furthermore, when

We can conclude that

where

This completes the proof Theorem 2.

The authors declare that they have no competing interests.

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

Afif Abdalmonem,Omer Abdalrhman,Shuangping Tao,1 1, (2016) Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz Spaces. Applied Mathematics,07,1165-1182. doi: 10.4236/am.2016.710104