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.


Introduction
is homogenous of degree zero on n  , The corresponding fractional maximal operator with variable kernel is defined by Especially, in the case 0 µ ≡ , 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 , T µ Ω was bounded from p L to q L .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 , , , ,  .Throughout this paper E denotes the Lebesgue measure, E χ means the characteristic function of a measurable set n S ⊂  .C always means a positive constant independent of the main parameters and may change from one occurrence to another.

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 n  with 0 E > .We first defined Lebesgue spaces with variable exponent.
is measurable : d for some constant 0 The Lebesgue spaces ⋅ is a Banach spaces with the norm defined by Let M be the Hardy-Littlewood maximal operator.We denote ( ) to be the set of all function and 0 λ ≤ < ∞ .The Herz-Morrey spaces with variable exponent  is defined by , , \ 0 : Remark 2.1.(See [6]) Comparing the Homogeneous Herz-Morrey Spaces with variable exponent with the homogeneous Herz spaces with variable exponent, where

Properties of Variable Exponent
In this section we state some properties of variable exponent belonging to the class ( ) ( ) Proposition 3.2.(see [13]) Suppose that ( ) ( ) ( ) ( ) 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, (See [16]) Suppose that n x ∈  , the variable function ( ) , then for all measurable function f and g, we have (See [17]) Suppose that ( ) ( ) and 0 p p 1) For any cube and 2 n Q ≤ , all the Q χ ∈ , then: 2) For any cube and 1 Q ≥ , then ( ) where ( ) , then there exist constant 0 C > such that for all balls B in n  and all measurable subset S R , there exist a constant 0 C > such that for any balls B in n  .we have

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 , and the integral modulus of continuity .
Below, we first estimate 1 U using size condition of j f .Minkowski inequality when 1 j k ≤ − , we get ( ) , , , , , , According to Lemma 3.4 and the formula ( ) ( ) . Combining Lemma 3.2, note that 1 q < ≤ , then we use the Hölder inequality and obtain Next we estimate 2 U , by using Proposition 3.2 we have ( ) First we estimate of 1 J , then we have To estimate 2 J , when λ α < , we have Complete prove Theorem A.
the above integral takes the Cauchy principal value.At this time 0 µ ≡ , , T µ Ω is much more close related to the elliptic partial equations of the second order with variable coefficients.Now we need the further assumption for condition if Ω meets the conditions i), ii) and the integral modulus of continuity of order r of Ω defined by