The Boundedness of Fractional Integral with Variable Kernel on Variable Exponent Herz-Morrey Spaces ()
Received 19 March 2016; accepted 24 April 2016; published 28 April 2016
![](//html.scirp.org/file/12-1720557x5.png)
1. Introduction
Let
,
is homogenous of degree zero on
,
denotes the unit sphere in
. If
i) For any
, one has
;
ii) ![](//html.scirp.org/file/12-1720557x13.png)
The fractional integral operator with variable kernel
is defined by
![](//html.scirp.org/file/12-1720557x16.png)
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
![](//html.scirp.org/file/12-1720557x21.png)
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).
NOTES
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*Corresponding author.