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Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 780132, 18 pages

http://dx.doi.org/10.1155/2012/780132

## A Weighted Variant of Riemann-Liouville Fractional Integrals on

^{1}Department of Mathematics, Linyi University, Shandong, Linyi 276005, China^{2}School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received 14 June 2012; Accepted 21 August 2012

Academic Editor: BashirΒ Ahmad

Copyright Β© 2012 Zun Wei Fu et al. 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 introduce certain type of weighted variant of Riemann-Liouville fractional integral on and obtain its sharp bounds on the central Morrey and -central BMO spaces. Moreover, we establish a sufficient and necessary condition of the weight functions so that commutators of weighted Hardy operators (with symbols in -central BMO space) are bounded on the central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Weyl and CesΓ ro.

#### 1. Introduction

Let . The well-known * Riemann-Liouville fractional integral * is defined by
for all locally integrable functions on . The study of Riemann-Liouville fractional integral has a very long history and number of papers involved its generalizations, variants, and applications. For the earlier development of this kind of integrals and many important applications in fractional calculus, we refer the interested reader to the book [1]. Among numerous material dealing with applications of fractional calculus to (ordinary or partial) differential equations, we choose to refer to [2] and references therein.

As the classical -dimensional generalization of , the well-known Riesz potential (the solution of Laplace equation) with is defined by setting, for all locally integrable functions on , where . The importance of Riesz potentials lies in the fact that they are indeed smoothing operators and have been extensively used in many different areas such as potential analysis, harmonic analysis, and partial differential equations. Here we refer to the paper [3], which is devoted to the sharp constant in the Hardy-Littlewood-Sobolev inequality related to .

This paper focused on another generalization, the weighted variants of Riemann-Liouville fractional integrals on . We investigate the boundedness of these weighted variants on the type of central Morrey and central Campanato spaces and also give the sharp estimates. This development begins with an equivalent definition of as More generally, we use a positive function (weight function) to replace in (1.3) and generalize the parameter from the positive axle to the Euclidean space therein. We then derive a weighted generalization of on , which is called the weighted Hardy operator (originally named weighted Hardy-Littlewood avarage) .

More precise, let be a positive function on . The * weighted Hardy operator * is defined by setting, for all complex-valued measurable functions on and ,
Under certain conditions on , Carton-Lebrun and Fosset [4] proved that maps , , into itself; moreover, the operator commutes with the Hilbert transform when , and with certain CalderΓ³n-Zygmund singular integrals including the Riesz transform when . Obviously, for and , if we take , then as mentioned above, for all ,

A further extension of [4] was due to Xiao [5] as follows.

Theorem A. *Let . Then, is bounded on if and only if
**
Moreover,
*

*Remark 1.1. *Notice that the condition (1.6) implies that is integrable on since . We naturally assume is integrable on throughout this paper.

Obviously, Theorem A implies the celebrated result of Hardy et al. [6, Theorem 329], namely, for all and ,
The constant in (1.6) also seems to be of interest as it equals to if and . In this case, is precisely reduced to the * classical Hardy operator * defined by
which is the most fundamental integral averaging operator in analysis. Also, a celebrated operator norm estimate due to Hardy et al. [6], that is,
with , can be deduced from Theorem A immediately.

Recall that is defined to be the space of all such that where and the supremum is taken over all balls in with sides parallel to the axes. It is well known that , since contains unbounded functions such as . Another interesting result of Xiao in [5] is that the weighted Hardy operator is bounded on , if and only if Moreover, In recent years, several authors have extended and considered the action of weighted Hardy operators on various spaces. We mention here, the work of Rim and Lee [7], Kuang [8], KruliΔ et al. [9], Tang and Zhai [10], Tang and Zhou [11].

The main purpose of this paper is to make precise the mapping properties of weighted Hardy operators on the central Morrey and -central BMO spaces. The study of the central Morrey and -central BMO spaces are traced to the work of Wiener [12, 13] on describing the behavior of a function at the infinity. The conditions he considered are related to appropriate weighted spaces. Beurling [14] extended this idea and defined a pair of dual Banach spaces and , where . To be precise, is a Banach algebra with respect to the convolution, expressed as a union of certain weighted spaces. The space is expressed as the intersection of the corresponding weighted spaces. Later, Feichtinger [15] observed that the space can be equivalently described by the set of all locally -integrable functions satisfying that where is the characteristic function of the unit ball , is the characteristic function of the annulus , , and is the norm in . By duality, the space , called Beurling algebra now, can be equivalently described by the set of all locally -integrable functions satisfying that Based on these, Chen and Lau [16] and GarcΓa-Cuerva [17] introduced an atomic space associated with the Beurling algebra and identified its dual as the space , which is defined to be the space of all locally -integrable functions satisfying that

By replacing with in (1.3) and (1.6), we obtain the spaces and , which are the homogeneous version of the spaces and , and the dual space of is just . Related to these homogeneous spaces, in [18, 19], Lu and Yang introduced the homogeneous counterparts of and , denoted by and , respectively. These spaces were originally denoted by and in [18, 19]. Recall that the space is defined to be the space of all locally -integrable functions satisfying that It was also proved by Lu and Yang that the dual space of is just .

In 2000, Alvarez et al. [20] introduced the following -central bounded mean oscillation spaces and the central Morrey spaces, respectively.

*Definition 1.2. * Let and . The central Morrey space is defined to be the space of all locally -integrable functions satisfying that

*Definition 1.3. * Let and . A function is said to belong to the -central bounded mean oscillation space if

We remark that if two functions which differ by a constant are regarded as a function in the space , then becomes a Banach space. Apparently, (1.19) is equivalent to the following condition:

*Remark 1.4. * is a Banach space which is continuously included in . One can easily check if , , , and if . Similar to the classical Morrey space, we only consider the case in this paper.

*Remark 1.5. *The space when is just the space . It is easy to see that for all . When , then the space is just the central version of the Lipschitz space .

*Remark 1.6. * If , then by HΓΆlder's inequality, we know that for , and for .

For more recent generalization about central Morrey and Campanato space, we refer to [21]. We also remark that in recent years, there exists an increasing interest in the study of Morrey-type spaces and the related theory of operators; see, for example, [22].

In this paper, we give sufficient and necessary conditions on the weight which ensure that the corresponding weighted Hardy operator is bounded on and . Meanwhile, we can work out the corresponding operator norms. Moreover, we establish a sufficient and necessary condition of the weight functions so that commutators of weighted Hardy operators (with symbols in central Campanato-type space) are bounded on the central Morrey-type spaces. These results are further used to prove sharp estimates of some inequalities due to Weyl and CesΓ ro.

#### 2. Sharp Estimates of

Let us state our main results.

Theorem 2.1. * Let and . Then is a bounded operator on if and only if
**
Moreover, when (2.1) holds, the operator norm of on is given by
*

*Proof. *Suppose (2.1) holds. For any , using Minkowski's inequality, we have
It implies that
Thus maps into itself.

The proof of the converse comes from a standard calculation. If is a bounded operator on , take
Then
where is the volume of the unit ball in .

We have
(2.8) together with (2.4) yields the desired result.

Corollary 2.2. *
(i) For , , and ,
**(ii) For and ,
*

Next, we state the corresponding conclusion for the space .

Theorem 2.3. * Let and . Then is a bounded operator on if and only if (2.1) holds. Moreover, when (2.1) holds, the operator norm of on is given by
*

*Proof. *Suppose (2.1) holds. If , then for any and ball , using Fubini's theorem, we see that

Using Minkowski's inequality, we have
which implies is bounded on and

Conversely, if is a bounded operator on , take
where and denote the right and the left halves of , separated by the hyperplane , and is the first coordinate of .

Thus, by a standard calculation, we see that and
From this formula we have

The proof is complete.

Corollary 2.4. *
(i) For and , we have
**(ii) For , we have .*

#### 3. A Characterization of Weight Functions via Commutators

A well-known result of Coifman et al. [23] states that the commutator generated by CalderΓ³n-Zygmund singular integrals and BMO functions is bounded on , . Recently, we introduced the commutators of weighted Hardy operators and BMO functions introduced in [24]. For any locally integrable function on and integrable function , the * commutator of the weighted Hardy operator * is defined by

It is easy to see that when and satisfies the condition (1.6), then the commutator is bounded on , . An interesting choice of is that it belongs to the class of . When symbols , the condition (1.6) on weight functions can not ensure the boundedness of on . Via controlling by the Hardy-Littlewood maximal operators instead of sharp maximal functions, we [24] established a sufficient and necessary (more stronger) condition on weight functions which ensures that is bounded on , where . More recently, Fu and Lu [25] studied the boundedness of on the classical Morrey spaces. Tang et al. [26] and Tang and Zhou [11] obtained the corresponding result on some Herz-type and Triebel-Lizorkin-type spaces. We also refer to the work [27] for more general -linear Hardy operators.

Similar to [24], we are devoted to the construction of a sufficient and necessary condition (which is stronger than in Theorem 2.1) on the weight functions so that commutators of weighted Hardy operators (with symbols in -central BMO space) are bounded on the central Morrey spaces. For the boundedness of commutators with symbols in central BMO spaces, we refer the interested reader to [28, 29] and Mo [30].

Theorem 3.1. * Let , . Assume further that is a positive integrable function on . Then, the commutator is bounded from to , for any , if and only if
*

*Remark 3.2. * The condition (2.1), that is, , is weaker than . In fact, let

By , we know that implies . But the following example shows that does not imply . For , if we take and , where , then and .

*Proof. *(i) Let . Denote by and by . Assume . We get

By the Minkowski inequality and the HΓΆlder inequality (with ), we have
Similarly, we have

Now we estimate ,
We see that
Therefore,

Combining the estimates of , , and , we conclude that is bounded from to .

Conversely, assume that for any , is bounded from to . We need to show that . Since , we will prove that and , respectively. To this end, let
for all . Then it follows from Remark 1.5 that , and

Let . Then

For , we obtain
So,
Therefore, we have

On the other hand,
since and are integrable functions on . Combining the above estimates, we get

Combining (3.18) and (3.16), we then obtain the desired result.

Notice that comparing with Theorems 2.1 and 2.3, we need a priori assumption in Theorem 3.1 that is integrable on . However, by Remark 1.1, this assumption is reasonable in some sense.

When with , namely, is a central -Lipschitz function, we have the following conclusion. The proof is similar to that of Theorem 3.1. We give some details here.

Theorem 3.3. * Let , , , , , and . If (2.1) holds true, then for all , the corresponding commutator is bounded from to . *

*Proof. *Let , , and be as in the proof of Theorem 3.1. Then, following the estimates of and in the proof of Theorem 3.1, we see that

For , we also have
Since now , we see that
Therefore,

Combining the estimates of , , and , we conclude that is bounded from to .

Different from Theorem 3.1, it is still unknown whether the condition (2.1) in Theorem 3.3 is sharp. That is, whether the fact that is bounded from to for all induces (2.1)?

More general, we may extend the previous results to the th order commutator of the weighted Hardy operator. Given and a vector , we define the higher order commutator of the weighted Hardy operator as When , we understand that . Notice that if , then .

Using the method in the proof of Theorems 3.1 and 3.3, we can also get the following Theorem 3.4. For the sake of convenience, we give the sketch of the proof of Theorem 3.4(i) here.

Theorem 3.4. *Let , , , , , , and .**
(i) Assume further that is a positive integrable function on . The commutator is bounded from to , for any , if and only if
**(ii) Let and . If (2.1) holds true, then the corresponding commutator is bounded from to . *

*Proof. *Let . Denote by and by . Assume . We get
Then, applying the Minkowski inequality and the HΓΆlder inequality (with ), and repeating the arguments in the proof of Theorem 3.1, is bounded from to for any , provided

Conversely, assume that is bounded from to for any . We choose with for all and . Then . Repeating the argument in the proof of Theorem 3.1 then yields the desired conclusion.

We point out that, it is still unknown whether the condition (2.1) in Theorem 3.4(ii) is sharp.

#### 4. Adjoint Operators and Related Results

In this section, we focus on the corresponding results for the adjoint operators of weighted Hardy operators.

Recall that the *weighted CesΓ ro operator * is defined by

If , , and , then is reduced to , where is a variant of Weyl integral operator and defined by
for all . When and , is the classical *CesΓ ro operator*:

It was pointed out in [5] that the weighted Hardy operator and the weighted CesΓ ro operator are adjoint mutually, namely, for all admissible pairs and .

Since and are a pair of dual Banach spaces, it follows from Theorem 2.1 the following.

Theorem 4.1. * Let . Then is bounded on if and only if
**
Moreover, when (4.5) holds, the operator norm of on is given by
*

Corollary 4.2. *
(i) For and ,
**(ii) For , we have
*

Since the dual space of is isomorphic to (see [18, 19]), Theorem 2.3 implies the following result.

Theorem 4.3. * Let . Then is a bounded operator on if and only if (4.5) holds. Moreover, when (4.5) holds, the operator norm of on is given by
*

Corollary 4.4. * For , we have
*

Following the idea in Section 3, we define the higher order commutator of the weighted CesΓ ro operator as When , is understood as . Notice that if , then . Similar to the proofs of Theorems 3.1 and 3.3, we have the following result.

Theorem 4.5. *Let , , , , , , and .**
(i) Assume further that is a positive integrable function on . The commutator is bounded from to , for any , if and only if
**
(ii) Let and . Then the corresponding commutator is bounded from to , provided that
*

We conclude this paper with some comments on the discrete version of the weighted Hardy and CesΓ ro operators.

Let be the set of all nonnegative integers and denote the set . Let now be a nonnegative function defined on and be a complex-valued measurable function on . The *discrete weighted Hardy operator * is defined by
and the corresponding *discrete weighted CesΓ ro operator* is defined by setting, for all ,
We remark that, by the same argument as above with slight modifications, all the results related to the operators and in Sections 1β4 are also true for their discrete versions and .

#### Acknowledgments

This work is partially supported by the Laboratory of Mathematics and Complex Systems, Ministry of Education of China and the National Natural Science Foundation (Grant nos. 10901076, 11101038, 11171345, and 10931001).

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