`Abstract and Applied AnalysisVolumeΒ 2012, Article IDΒ 929381, 14 pageshttp://dx.doi.org/10.1155/2012/929381`
Research Article

## Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 20 March 2012; Revised 5 July 2012; Accepted 20 July 2012

Copyright Β© 2012 Zengyan Si and Fayou Zhao. 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

We prove that is in if and only if the commutator of the multiplication operator by and the general fractional integral operator is bounded from the weighted Morrey space to , where , , , , and , and here denotes the critical index of for the reverse HΓΆlder condition.

#### 1. Introduction and Main Results

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 . Since we assume only upper bound on heat kernel and no regularity on its space variables, this property (1.1) is satisfied by a class of differential operator, see [1] for details.

For , the general 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. Let be a locally integrable function on , the commutator of and is defined by

For the special case of , many results have been produced. PaluszyΕski [2] obtained that if the commutator is bounded from to , where and with . Shirai [3] proved that if and only if the commutator is bounded from the classical Morrey spaces to for , and or to for , and . Wang [4] established some weighted boundedness of properties of commutator on the weighted Morrey spaces under appropriated conditions on the weight , where the symbol belongs to (weighted) Lipschitz spaces. The weighted Morrey space was first introduced by Komori and Shirai [5]. For the general case, Wang [6] proved that if , then the commutator is bounded from to , where , and .

The purpose of this paper is to give necessary and sufficient conditions for boundedness of commutators of the general fractional integrals with (the weighted Lipschitz space). Our theorems are the following.

Theorem 1.1. Let , , , and . Then one has the following. (a)If , then is bounded from to ; (b)If is bounded from to , then .

Theorem 1.2. Let , , , and , where denotes the critical index of for the reverse HΓΆlder condition. Then one has the following. (a)If , then is bounded from to ; (b)If is bounded from to , then .

Our results not only extend the results of [4] from to a general operator , but also characterize the (weighted) Lipschitz spaces by means of the boundedness of on the weighted Morrey spaces, which extend the results of [4, 6]. The basic tool is based on a modification of sharp maximal function introduced by [7].

Throughout this paper all notation is standard or will be defined as needed. Denote the Lebesgue measure of by and the weighted measure of by , where . For a measurable set , denote by the characteristic function of . For a real number , , let be the dual of such that . The letter will be used for various constants, and may change from one occurrence to another.

#### 2. Some Preliminaries

A nonnegative function defined on is called weight if it is locally integral. A weight is said to belong to the Muckenhoupt class for , if there exists a positive constant such that for every ball . The class is defined replacing the above inequality by When , if there exist positive constants and such that given a ball and is a measurable subset of , then

A weight function belongs to for if for every ball in , there exists a positive constant which is independent of such that

From the definition of , we can get that Since , then by (2.5), we have .

A weight function belongs to the reverse HΓΆlder class if there exist two constants and such that the following reverse HΓΆlder inequality, holds for every ball in .

It is well known that if with , then there exists such that . It follows from HΓΆlder inequality that implies for all . Moreover, if , then we have for some . We thus write to denote the critical index of for the reverse HΓΆlder condition. For more details on Muckenhoupt class , we refer the reader to [8β10].

Definition 2.1 (see [5]). 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 .

Definition 2.2 (see [11]). Let , , and . A locally integral function is said to be in if where and the supremum is taken over all ball . When , we denote by .

Obviously, for the case , then the space is the classical space.

Remark 2.3. Let , GarcΓ­a-Cuerva [11] proved that the spaces coincide, and the norms of are equivalent with respect to different values of provided that .

Given a locally integrable function and , , define the fractional maximal function by when . If and , then denotes the usual Hardy-Littlewood maximal function.

Let be a weight. The weighted maximal operator is defined by The fractional weighted maximal operator is defined by where and . For any , the sharp maximal function associated the generalized approximations to the identity is given by Martell [7] as follows: where and is the radius of the ball . For , we introduce the -sharp maximal operator as which is a modification of the sharp maximal operator of Stein and Murphy [9]. Set . Using the same methods as those of [9, 12], we can get the following.

Lemma 2.4. Assume that the semigroup has a kernel which satisfies the upper bound (1.1). Let and for some . Suppose that , then for every , there exists a real number independent of such that one has the following weighted version of the local good inequality, for , , where is a fixed constant which depends only on .

If , then In particular, when and , we have

#### 3. Proof of Theorem 1.1

To prove Theorem 1.1, we need the following lemmas.

Lemma 3.1 (see [1]). Assume that the semigroup has a kernel which satisfies the upper bound (1.1). Then for , the difference operator has an associated kernel which satisfies

Lemma 3.2 (see [4]). Let , , and . Then for every and , one has

Lemma 3.3 (see [5]). Let , , and . Then for every , one has

Lemma 3.4 (see [4]). Let , , , and . Then for every , one has

Lemma 3.5. Let , , , and . Then for every , one has

Proof. Since the semigroup has a kernel which satisfies the upper bound (1.1), it is easy to check that for all . Together with the result (cf. [4]), that is, we can get the desired result.

Remark 3.6. Using the boundedness property of , we also know is bounded from to weak . It is easy to check that Lemmas 3.2β3.5 also hold when .

The following lemma plays an important role in the proof of Theorem 1.1.

Lemma 3.7. Let , and . Then for all and for all , one has

The same method of proof as that of Lemma 4.7 (see below), one omits the details.

Proof of Theorem 1.1. We first prove . We only prove Theorem 1.1 in the case . For the general case , the method is the same as that of [1]. We omit the details.
For and , we can find a number such that . By (2.17) and Lemma 3.7, we obtain the following: Let and . Since , then by (2.5), we have . Since , by Lemmas 3.2β3.5, we yield that
Now we prove . Let be the Laplacian on , then is the classical fractional integral . Let and weight , then and . From [2], the boundedness of implies that .
Thus Theorem 1.1 is proved.

#### 4. Proof of Theorem 1.2

We also need some Lemmas to prove Theorem 1.2.

Lemma 4.1 (see [4]). Let , and . Then if and , one has

Lemma 4.2 (see [4]). Let , and . Then if and , one has

Lemma 4.3 (see [4]). Let , , and . For any , one has

Lemma 4.4. Let , and . Then if and , one has

Proof. As before, we know that for all . Using the boundedness property of on weighted Morrey space (cf. [4]), we have where and .

Remark 4.5. It is easy to check that the above lemmas also hold for .

Lemma 4.6. Assume that the semigroup has a kernel which satisfies the upper bound (1.1), and let . Then, for every function , and , one has

Proof. Fix and . Then, It follows from and that Thus, HΓΆlder's inequality and Definition 2.2 lead to the following: Moreover, for any and , we have and We will estimate the values of terms and , respectively.
Using HΓΆlder's inequality and Remark 2.3, we have the following:
By a simple calculation, we have Since , by the HΓΆlder inequality, we get
Thus, Lemma 4.6 is proved.

Lemma 4.7. 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 are going to estimate each term, respectively. Fix and choose a real number such that and . Since , then it follows from HΓΆlder's inequality that

For , using HΓΆlder's inequality, Kolmogorov's inequality (see page 485 [8],) and Remark 3.6, then we deduce that where we have used the condition that .

Using HΓΆlder's inequality and Lemma 4.6, we obtain that

For , using the estimate in II, we get

By virtue of Lemma 3.1, we have the following: Making use of the same argument as that of , we have

Note that , So, the value of can be controlled by

Combining the above estimates for IβV, we finish the proof of Lemma 4.7.

Proof of Theorem 1.2. We first prove . As before, we only prove Theorem 1.2 in the case . For and , we can find a number such that . By Lemma 4.7, we obtain the following: Let and . Lemmas 4.1β4.4 yield that
Now we prove (b). Let be the Laplacian on , then is the classical fractional integral . We use the same argument as Janson [13]. Choose so that . For can be written as the absolutely convergent Fourier series, with since . For any and , let and , where . Fix and , then , hence, This implies that . Thus, is proved.

#### Acknowledgment

The authors thank the referee for the useful suggestions. Z. Si was supported by Doctoral Foundation of Henan Polytechnic University. F. Zhao was supported by Shanghai Leading Academic Discipline Project (Grant no. J50101).

#### References

1. X. T. Duong and L. X. Yan, βOn commutators of fractional integrals,β Proceedings of the American Mathematical Society, vol. 132, no. 12, pp. 3549β3557, 2004.
2. M. Paluszyński, βCharacterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss,β Indiana University Mathematics Journal, vol. 44, no. 1, pp. 1β17, 1995.
3. S. Shirai, βNecessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces,β Hokkaido Mathematical Journal, vol. 35, no. 3, pp. 683β696, 2006.
4. H. Wang, βOn some commutator theorems for fractional integral operators on the weighted morrey spaces,β http://128.84.158.119/abs/1010.2638v1.
5. Y. Komori and S. Shirai, βWeighted Morrey spaces and a singular integral operator,β Mathematische Nachrichten, vol. 282, no. 2, pp. 219β231, 2009.
6. H. Wang, βSome estimates for the commutators of fractional integrals associated to operators with Gaussian kenerl bounds,β http://xxx.tau.ac.il/abs/1102.4380v1.
7. J. M. Martell, βSharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications,β Studia Mathematica, vol. 161, no. 2, pp. 113β145, 2004.
8. J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam, The Netherlands, 1985.
9. E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Monographs in Harmonic Analysis, Princeton University Press, Princeton, NJ, USA, 1993.
10. A. Torchinsky, Real-Variable Methods in Harmonic Analysis, vol. 123, Academic Press, Orlando, Fla, USA, 1986.
11. J. García-Cuerva, βWeighted ${H}^{p}$ spaces,β Dissertations Math, vol. 162, pp. 1β63, 1979.
12. C. Pérez, βEndpoint estimates for commutators of singular integral operators,β Journal of Functional Analysis, vol. 128, no. 1, pp. 163β185, 1995.
13. S. Janson, βMean oscillation and commutators of singular integral operators,β Arkiv för Matematik, vol. 16, no. 2, pp. 263β270, 1978.