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

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*Applied Mathematics*,

**7**, 1165-1182. doi: 10.4236/am.2016.710104.

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

1. Introduction

Let, is homogenous of degree zero on, denotes the unit sphere in. If

(i) For any, one has;

(ii)

The fractional integral operator with variable kernel is defined by

The commutators of the fractional integral is defined by

When, the above integral takes the Cauchy principal value. At this time, is much more close related to the elliptic partial equations of the second order with variable coefficients. Now we need the further assumption for. It satisfies

For, we say Kernel function satisfies the -Dini condition, if meets the conditions (i), (ii) and

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

where is the a rotation in

when, is the fraction integral operator

The corresponding fractional maximal operator with variable kernel is defined by

We can easily find that when is just the fractional maximal operator

Especially, in the case, 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 was bounded from to. 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 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

and their commutators are bonuded on Herz spaces with two variable exponents or not.

Throughout this paper denotes the Lebesgue measure, means he characteristic function of a measurable set. 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 with. We first define the Lebesgue spaces with variable exponent.

Definition 2.1. see [1] Let be a measurable function. The Lebesgue space with variable

exponent is defined by

The space is defined by

The Lebesgue spaces is a Banach spaces with the norm defined by

We denote

.

Then consists of all satisfying and.

Let M be the Hardy-Littlewood maximal operator. We denote to be the set of all function satisfying the M is bounded on.

Definition 2.2. see [17] Let. The mixed Lebesgue sequence space with variable exponent

is the collection of all sequences of the measurable functions on such that

Noticing, we see that

Let

Definition 2.3. see [16] Let. The homogeneous Herz space with variable ex- ponent is defined by

where

Remark 2.1. see [16] (1) If satisfying, then

(2) If and, then and. Thus, by Lemma 3.7

and Remark 2.2, for any, we have

where

This implies that.

Remark 2.2. Let. then

where

Definition 2.4. see [18] For, the Lipschitz space is defined by

(1.1)

3. Properties of Variable Exponent

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

Proposition 3.1. see [1] If satisfies

then, we have.

Proposition 3.2. see [15] Suppose that,. Let, and define the variable exponent by:. Then we have that for all,

Proposition 3.3. Suppose that, , ,. Let

, and define the variable exponent by:. Then

Proof

By Proposition 3.2, we get

Now, we need recall some lemmas

Lemma 3.1. see [13] Given have that for all function f and g,

Lemma 3.2. see [19] Suppose that, , satisfies the -Dini con- dition. If there exists an such that then

Lemma 3.3. see [20] Suppose that, the variable function is defined by,

then for all measurable function f and g, we have

Lemma 3.4. see [21] Suppose that and.

1) For any cube and, all the, then:

2) For any cube and, then where

Lemma 3.5. see [22] If, then there exist constants such that for all balls B in and all measurable subset

Lemma 3.6. see [13] If, there exist a constant such that for any balls B in. we have

Lemma 3.7. see [16] Let. If, then

4. Main Theorems and Their Proof

Theorem 1. Suppose that, , ,

with. And let satisfy and define the vari-

able exponent by. Then the operators is bounded from to

.

Theorem 2. Let Suppose that,

, with. If satisfy

and define the variable exponent by. Then the com-

mutators is bounded from to.

Proof of Theorem1:

Let. We write

From definition of

Since

where

And, thus

That is

This implies only to prove. Denote

Now we consider. Applying Lemma 3.7

where

By the Proposition 3.2, we get

Since, then we have, and

By Lemma 3.7 and Remark 2.2, we get

Hence, and, this implies that

Now, we estimate of using size condition of and Minkowski inequality, when we get,

Since we define the variable exponent, by Lemma 3.3 we get

According Lemma 3.4 and the formula, then we have

(1.2)

By Lemma 3.2, we get

It follows that

(1.3)

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

where

Since, then we have, and

Now if, then we have

where

If, then we have

where, this implies that

Finally, we estimate by Lemma 3.7, we get

(1.4)

Note that, when, , then. Therefore, applying the generalized Hölder’s In- equality, we have

Define the variable exponent by Lemma 3.3, then we have

According Lemma 3.4 and the formula, we have Then we get

(1.5)

From Equations (1.4), (1.5) and using Lemma 3.7, and we can obtain

Note that

see [9] .

Then we have

where

Since and, as the same we have

This completes the proof Theorem 1.

Proof of Theorem 2

Let,. We write

From definition of

Since

where

And. The similar to prove of Theorem 1

Hence. Denote

First we estimate. Note that is bonuded on (Proposition 3.3), similarly to esti-

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

That is

Now, we estimate of. Using size condition of and Minkowski inequality, when we get,

We have that

(1.6)

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

(1.7)

By (1.7) and lemma 3.7, we obtain that

where

Since and, the similar way to estimate in the proof of Theorem1, we can obtain that

where, this implies that

Finally, we estimate. Note that, when, , then, we can obtain that

Then we have

(1.8)

Applying the generalized Hölder’s Inequality, we get

Define the variable exponent by Lemma 3.3, then we have

According Lemma 3.4 and the formula, we have. Then we get

By (1.8), we can obtain that

(1.9)

Then by (1.9) and Lemma 3.7, we have

where

Furthermore, when, note that see [9] , the similar way to estimate, we get

We can conclude that

where, this implies that

This completes the proof Theorem 2.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgements

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

NOTES

^{*}Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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