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

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.

• Fractional Integral
• Variable Kernel
• Variable Exponent
• Herz-Morrey Spaces

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

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-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 was bounded from to. 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 to. Throughout this paper denotes the Lebesgue measure,

means the 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 Lebesgue spaces and Herz-Morrey spaces with variable exponent.

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

Definition 2.1. 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.

Let

Definition 2.2. Let and. The Herz- Morrey spaces with variable exponent is defined by

Remark 2.1. (See [6] ) Comparing the Homogeneous Herz-Morrey Spaces with variable exponent with the homogeneous Herz spaces with variable exponent, where is defined by

Obviously,

3. Properties of Variable Exponent

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

Proposition 3.1. (See [12] ) If satisfies

then, we have.

Proposition 3.2. (see [13] ) Suppose that,. Let, and define the variable exponent by:. Then we have that for all,

Now, we need recall some lemmas

Lemma 3.1. (See [14] ) Given have that for all function f and g,

Lemma 3.2. (See [15] ) Suppose that, , satisfies the -Dini condition. If there exists an such that then

Lemma 3.3. (See [16] ) Suppose that, the variable function is defined by,

then for all measurable function f and g, we have

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

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

2) For any cube and, then where

Lemma 3.5. (See [18] ) If, then there exist constant such that for all balls B in and all measurable subset

such that is constants satisfying

Lemma 3.6. (See [14] ) If, there exist a constant such that for any balls B in. we have

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. Let

, and the integral modulus of continuity satisfies

And let satisfy and define the variable exponent by , then we have

For all

Proof If arbitrarily, we apply inequality

If we denote

Then we have

Below, we first estimate using size condition of. Minkowski inequality when, we get

Then we have

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

According to Lemma 3.4 and the formula, then we have

. Combining Lemma 3.2, note that, we get

It follows that

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

Hence we have

Remark that. We consider the two cases and. If, then we use the Hölder inequality and obtain

If, , we get

Next we estimate, by using Proposition 3.2 we have

First we estimate of, then we have

To estimate of, when, we have

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).

References

1. Kenig, C. (1994) Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. American Mathematical Society, Providence. http://dx.doi.org/10.1090/cbms/083

2. Calderón, A. and Zygmund, A. (1955) On a Problem of Mihilim. Transactions of the American Mathematical Society, 78, 209-224. http://dx.doi.org/10.2307/1992955

3. Calderón, A. and Zygmund, A. (1978) On Singular Integral with Variable Kernels. Journal of Applied Analysis, 7, 221-238.

4. Christ, M., Duoandikoetxea, J. and Rubio de Francia, J. (1986) Maximal Operators Related to the Radon Transform and the Calderóon-Zygmund Method of Rotations. Duke Mathematical Journal, 53, 189-209. http://dx.doi.org/10.1215/S0012-7094-86-05313-5

5. Muckenhoupt, B. and Wheeden, R. (1971) Weighted Norm Inequalities for Singular and Fractional Integrals. Transactions of the American Mathematical Society, 161, 249-258. http://dx.doi.org/10.1090/S0002-9947-1971-0285938-7

6. Izuki, M. (2010) Boundedness of Commutators on Herz Spaces with Variable Exponent. Rendiconti del Circolo Matematico di Palermo, 59, 199-213. http://dx.doi.org/10.1007/s12215-010-0015-1

7. Izuki, M. (2010) Fractional Integrals on Herz-Morrey Spaces with Variable Exponent. Hiroshima Mathematical Journal, 40, 343-355,.

8. Izuki M. ,et al. (2009)Boundedness of Vector-Valued Sublinear Operators on Herz-Morrey Spaces with Variable Exponent Mathematical Sciences Research Journal 13, 243-253.

9. Lu, S. and Xu, L. (2005) Boundedness of Rough Singular Integral Operators on the Homogeneous Morrey-Herz Spaces. Hokkaido Mathematical Journa, 34, 299-314. http://dx.doi.org/10.14492/hokmj/1285766224

10. Izuki M. ,et al. (2009)Herz and Amalgam Spaces with Variable Exponent, the Haar Wavelets and Greediness of the Wavelet System East Journal on Approximations 15, 87-109.

11. Wang, Z. and Zhu, Y. (2014) Boundedness of Multilinear Fractional Integral Operators on Herz-Morrey Spaces with Variable Exponent. Journal of Lantong University (Natural Science Edition), 13, 60-68.

12. Cruz-Uribe, D. and Fiorenza, A. (2013) Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Springer, New York.

13. Wu, H. and Lan, J. (2012) The Boundedness for a Class of Rough Fractional Integral Operators on Variable Exponent Lebesgue Spaces. Analysis in Theory and Applications, 28, 286-293.

14. Tan, J. and Liu, Z. (2015) Some Boundedness of Homogeneous Fractional Integrals on Variable Exponent Function Spaces. ACTA Mathematics Science (Chinese Series), 58, 310-320.

15. Cruz-Uribe, D., Fiorenza, A., Martell, J. and Pe’rez, C. (2006) The Boundedness of Classical Operators on Variable Lp Spaces. Annales Academiae Scientiarum Fennicae Mathematica, 31, 239-264.

16. Diening, L., Harjulehto, P., H&aumlst&ouml, P. and Ruziccka, M. (2011) Lebesgue and Sobolev Spaces with Variable Exponents. Springer-Verlag, Berlin Heidelberg. http://dx.doi.org/10.1007/978-3-642-18363-8

17. Ding, Y. and Lu, S. (2000) Homogeneous Fractional Integrals on Hardy Spaces. Tohoku Mathematical Journal, 52, 153-162.

18. Izuki, M. (2010) Boundedness of Sublinear Operators on Herz Spaces with Variable Exponent and Application to Wavelet Characterization. Analysis Mathematica, 36, 33-50. http://dx.doi.org/10.1007/s10476-010-0102-8