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