Received 19 March 2016; accepted 24 April 2016; published 28 April 2016
Let, is homogenous of degree zero on, denotes the unit sphere in. If
i) For any, one has;
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  -  . In 1971, Muckenhoupt and Wheeden  had proved the operator was bounded from to. In 1991, Kováčik and Rákosník  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  , and the boundedness of vector-valued sub-linear operator and fractional integral on Herz-Morrey spaces with variable exponent was introduced by authors  and  . We also note that Herz-Morrey spaces with variable exponent are generalization of Morrey-Herz spaces  and Herz spaces with variable exponent  . Recently, Wang Zijian and Zhu Yueping  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
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. Let and. The Herz- Morrey spaces with variable exponent is defined by
Remark 2.1. (See  ) Comparing the Homogeneous Herz-Morrey Spaces with variable exponent with the homogeneous Herz spaces with variable exponent, where is defined by
3. Properties of Variable Exponent
In this section we state some properties of variable exponent belonging to the class and .
Proposition 3.1. (See  ) If satisfies
then, we have.
Proposition 3.2. (see  ) 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  ) Given have that for all function f and g,
Lemma 3.2. (See  ) Suppose that, , satisfies the -Dini condition. If there exists an such that then
Lemma 3.3. (See  ) Suppose that, the variable function is defined by,
then for all measurable function f and g, we have
Lemma 3.4. (See  ) Suppose that and.
1) For any cube and, all the, then:
2) For any cube and, then where
Lemma 3.5. (See  ) 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  ) 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
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.
The authors declare that they have no competing interests.
This paper is supported by National Natural Foundation of China (Grant No. 11561062).
 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.
 Muckenhoupt, B. and Wheeden, R. (1971) Weighted Norm Inequalities for Singular and Fractional Integrals. Transactions of the American Mathematical Society, 161, 249-258.
 Lu, S. and Xu, L. (2005) Boundedness of Rough Singular Integral Operators on the Homogeneous Morrey-Herz Spaces. Hokkaido Mathematical Journa, 34, 299-314.
 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.
 Diening, L., Harjulehto, P., Hästö, P. and Ruziccka, M. (2011) Lebesgue and Sobolev Spaces with Variable Exponents. Springer-Verlag, Berlin Heidelberg.
 Izuki, M. (2010) Boundedness of Sublinear Operators on Herz Spaces with Variable Exponent and Application to Wavelet Characterization. Analysis Mathematica, 36, 33-50.