Boundedness of Fractional Integral with Variable Kernel and Their Commutators on Variable Exponent Herz Spaces ()
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
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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
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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
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From definition of ![]()
![]()
Since
![]()
where
![]()
![]()
![]()
And
, thus
![]()
That is
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This implies only to prove
. Denote ![]()
Now we consider
. Applying Lemma 3.7
![]()
where
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By the Proposition 3.2, we get
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Since
, then we have
, and
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By Lemma 3.7 and Remark 2.2, we get
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Hence
, and
, this implies that
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Now, we estimate of
using size condition of
and Minkowski inequality, when
we get,
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Since
we define the variable exponent
, by Lemma 3.3 we get
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According Lemma 3.4 and the formula
, then we have
(1.2)
By Lemma 3.2, we get
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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
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Since
, then we have
, and
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Now if
, then we have
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where ![]()
If
, then we have
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where
, this implies that
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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
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Note that
see [9] .
Then we have
![]()
where
![]()
Since
and
, as the same
we have
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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
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By (1.8), we can obtain that
(1.9)
Then by (1.9) and Lemma 3.7, we have
![]()
where
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Furthermore, when
, note that
see [9] , the similar way to estimate
, we get
![]()
We can conclude that
![]()
where
, this implies that
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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
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*Corresponding author.