Abstract and Applied Analysis

Volume 2014, Article ID 413716, 8 pages

http://dx.doi.org/10.1155/2014/413716

## Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

^{1}School of Computer Science and Technique, Henan Polytechnic University, Jiaozuo 454000, China^{2}School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

Received 10 December 2013; Accepted 18 May 2014; Published 1 June 2014

Academic Editor: Guozhen Lu

Copyright © 2014 Zhiheng Wang and Zengyan Si. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds, and let be the fractional integrals of for . For any locally integrable function , the commutators associated with are defined by . When (weighted space) or , the authors obtain the necessary and sufficient conditions for the boundedness of on weighted Morrey spaces, respectively.

#### 1. Introduction and Main Results

Morrey [1] introduced the classical Morrey spaces to investigate the local behavior of solutions to second order elliptic partial differential equations. Chiarenza and Frasca [2] established the boundedness of the Hardy-Littlewood maximal operator, the fractional operator, and a singular integral operator on the Morrey spaces. On the other hand, Coifman and Fefferman [3] and Muckenhoupt [4] studied the boundedness of these operators on weighted spaces. Motivated by these works, Komori and Shirai [5] introduced the following weighted Morrey space and investigated the boundedness of classical operators in harmonic analysis, that is, the Hardy-Littlewood maximal operator, a Calderón-Zygmund operator, the fractional integral operator, and so forth.

Let and . Then for two weights and , the weighted Morrey space is defined by where and the supremum is taken over all balls in .

If , then we have the classical Morrey space with measure . When , then is the Lebesgue space with measure .

Suppose that is a linear operator on which generates an analytic semigroup with a kernel satisfying a Gaussian upper bound, that is, for and all .

For , the fractional integral of the operator is defined by Note that if is the Laplacian on , then is the classical fractional integral which plays important roles in many fields. It is well known that is bounded from to for all , and is also of weak type .

Let and be a weight function. A locally integrable function is said to be in if where and the supremum is taken over all balls .

Let ; García-Cuerva [6] proved that the spaces coincide, and the norms of are equivalent with respect to different values provided that .

Let be a locally integrable function on ; we consider the commutator defined by

Chanillo [7] proved that the commutator of the multiplication operator by is bounded on for .

Duong and Yan [8] proved that is bounded from to , where , , , .

Mo and Lu [9] proved that the multilinear commutator generated by and is bounded from to , where , , for .

Lu et al. [10] proved that is bounded from to if and only if .

Wang [11] proved that is bounded from to , where , , and .

Inspired by the above results, we study the boundedness properties of the commutator on weighted Morrey spaces in this work. The main theorems are stated as follows.

Theorem 1. *Let , and , where denotes the critical index of for the reverse Hölder condition. Then the following conditions are equivalent. *(a)*. *(b)* is bounded from to .*

In particular, when in Theorem 1, we get the following.

Corollary 2. *Let , and , where denotes the critical index of for the reverse Hölder condition. Then the following conditions are equivalent. *(a)*. *(b)* is bounded from to .**Furthermore, if is the Laplacian, then the following conditions are equivalent. **. ** is bounded from to .*

Theorem 3. *Let , and such that , . Then the following conditions are equivalent. *(a)*. *(b)* is bounded from to .*

In particular, when in Theorem 3, we obtain the following.

Corollary 4. *Let , , and , such that , . Then the following conditions are equivalent. *(a)*. *(b)* is bounded from to .**Furthermore, if is the Laplacian, then the following conditions are equivalent. **. ** is bounded from to .*

*Remark 5. *It is easy to see that our results extend the results in [7, 8, 10, 11] significantly.

#### 2. Prerequisite Material

Let us first recall some definitions.

*Definition 6. *The Hardy-Littlewood maximal operator is defined by
Let be a weight. The weighted maximal operator is defined by
A variant of this maximal operator will become the main tool in our scheme; for ,
For , the fractional maximal operator is defined by
and the fractional weighted maximal operator is defined by
For any , the sharp maximal function associated with the generalized approximations to the identity is given by
where and is the radius of the ball .

In the above definitions, the supremum is taken over all balls containing .

*Definition 7. *A weight function is said to be in the Muckenhoupt class with if, for every ball in , there exists a positive constant which is independent of such that
When , if

When , if there exist positive constants and such that, given a ball and a measurable subset of ,

*Definition 8. *A weight function belongs to for if, for every ball in , there exists a positive constant which is independent of such that
where denotes the conjugate exponent of , that is, .

*Definition 9. *A weight function belongs to the reverse Hölder class if there exist two constants and such that the reverse Hölder inequality
holds for every ball in .

It is well known that if with , then there exists such that . It follows from Hölder's inequality that implies for all . Moreover, if , then we have for some . We thus write to denote the critical index of for the reverse Hölder condition.

We will make use of the following lemmas. We first provide a weighted version of the local good inequality for which allows us to obtain an analog of the classical Fefferman-Stein (see [3, 12]) estimate on weighted Morrey spaces.

Lemma 10 (see [13]). *Assume that the semigroup has a kernel which satisfies the upper bound (3). Take , and a ball such that there exists with . Then, for every , one can find (independent of ) and constant (which only depend on ), such that
**
where is a fixed constant which depends only on .*

As a consequence, by using the standard arguments, we have the following estimates.

For every , with . if , then In particular, when and , we have

Lemma 11 (see [11]). *Let , , and . Then if and , one has
**
The same conclusion still holds for .*

Lemma 12 (see [11]). *Let , and . Then if , and , one has
*

Lemma 13 (see [11]). *Consider , , and . For any , one has
*

*Remark 14. *By checking the proof of Lemmas 11–13, we know that the three lemmas above still hold when .

Lemma 15. *Let , and . Then if and , one has
*

*Proof. *Since the semigroup has a kernel which satisfies the upper bound (3), it is easy to check that for all . Using the boundedness property of on weighted Morrey space (see Lemma 11), we have
where and .

*Remark 16. *Since is of weak type . from the proof of Lemma 15, we can get that is also of weak type .

Lemma 17 (see [8, 14]). *Assume that the semigroup has a kernel which satisfies the upper bound (3). Then for , the difference operator has an associated kernel which satisfies
**
for some positive constant .*

Lemma 18. *Assume that the semigroup has a kernel which satisfies the upper bound (3), and let . Then, for every function , and for all , one has
**
where being the radius of .*

*Proof. *For any and . We have
For any and . We have
Thus,
Moreover, for any and , we have and :
We estimate each term in turn. For , we apply Hölder's inequalities with exponent . Then we have

Since , then . This fact together with Hölder's inequality impliesThen Lemma 18 is proved.

Lemma 19. *Let , and . Then for all and for all , one has
*

*Proof. *For any given , fix a ball which contains . We decompose , where . Observe that
Then
We estimate each term separately.

Since , then it follows from Hölder's inequality that

Applying Kolmogorov's inequality (see [15, page 485]), Hölder's inequality, and the continuity of , we thus have
By Lemma 18, we have
For , using the estimate obtained in , we get

By virtue of Lemma 17, we have

For , applying the same arguments as in , we get

Since , then . Thus,
Then
Combining the above estimates , we get (34). The proof of Lemma 19 is complete.

#### 3. Proofs of the Main Results

In this section we prove our main results. We start with the proof of Theorem 1.

*Proof. *(a)(b): Applying Lemmas 10 and 19, we get

Since , and , by making use of Lemmas 11–13, then we obtain
The last inequality follows from Lemma 15. This completes the proof of (a)(b).

(b)(a): Let be the Laplacian on ; then is the classical fractional integral . Choose so that . For , can be written as the absolutely convergent Fourier series with since . For any and , let and ,where . Fix and and we have ; hence, we have

This implies . Thus Theorem 1 is proved.

Similarly, to prove Theorem 3, we need the following lemmas.

Lemma 20. *Let such that and . Then for all and for all , one has
**
where .*

*Proof. *The case was proved by Duong and Yan (see [8] for details). The general case follows by repeating the same steps as in Lemma 19. Since the main steps and the ideas are almost the same, here we omit the proof.

Lemma 21 (see [5]). *If , and , then the fractional maximal operator is bounded from to .*

Lemma 22 (see [5]). *If , and , then the fractional maximal operator is bounded from to .*

Lemma 23 (see [5]). *If , and , then is bounded on .*

*Remark 24. *By applying the same argument as in Lemma 15, we know that the conclusion in Lemma 22 still holds for . We omit the proof here.

*Remark 25. *By checking the proof of Lemmas 21–23, we know that the three lemmas above still hold when .

Now we prove Theorem 3.

*Proof. *(a)(b): Since , then we get and . Applying Lemmas 10 and 20–23, we getIn the last inequality, we used the fact that is bounded from to (see Remark 24).

(b)(a): Let be the Laplacian on ; then is the classical fractional integral . Let and weight , and then and . From [10] we deduce that the boundedness of implies . Thus Theorem 3 is proved.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11226102) and the Doctoral Foundation of Henan Polytechnic University (no. B2012-055).

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