Endpoint Estimates for Fractional Hardy Operators and Their Commutators on Hardy Spaces

Abstract

bounds of fractional Hardy operators are obtained. Moreover, the estimates for commutators of fractional Hardy operators on Hardy spaces are worked out. It is also proved that the commutators of fractional Hardy operators are mapped from the Herz-type Hardy spaces into the Herz spaces. The estimates for multilinear commutators of fractional Hardy operators are also discussed.

1. Introduction

The most fundamental averaging operator is the Hardy operators defined by where the function is nonnegative integrable on and . A classical inequality, due to Hardy et al. [1], states that holds for and the constant is the best possible.

The Hardy integral inequality has received considerable attention. A number of papers involved its alternative proofs, generalizations, variants, and applications. Among numerous papers dealing with such inequalities, we choose to refer to the papers [2–4].

Let be a locally integrable function on ,  . In [5], Fu et al. defined -dimensional fractional Hardy operators : When , is just -dimensional Hardy operators ; see [3] for more details.

It is well known that averaging operators play an important role in harmonic analysis. For example, the Hardy-Littlewood maximal operators control many kinds of operators in analysis. Therefore, the study of Hardy operators is meaningful and has been fully discussed (see [4, 6–11]). In 2012, Zhao et al. [11] have shown the following results: when ,   maps into ;      maps into was false;     maps . Furthermore, (3) is also true for , . Recently, fractional Hardy operators were studied by many authors (see [5, 12, 13]). In 2007, Fu et al. [5] have obtained that the commutator is bounded from to , where . In 2013, Lu et al. [13] have proved that fractional Hardy operators map into , where , , and . Inspired by the above, we consider the endpoint estimates for fractional Hardy operators and their commutators on Hardy-type spaces.

This paper is organized as follows. In the second section, we give the (, ) bounds of fractional Hardy operators. In the third section, we obtain the estimates for commutators of fractional Hardy operators on Hardy spaces. In Section 4, we consider the case on Herz-type Hardy spaces. In Section 5, we obtain the estimate for multilinear commutators of fractional Hardy operators.

Throughout this paper, denotes that is equivalent to , which means there exist two positive constants and such that . For , and , and are conjugate indices; that is, .

Let us introduce some definitions below.

Definition 1. Let and . One says that if and only if where . The norm of is defined by

Remark 2. When , ; when , . We choose to refer to papers [5, 9].

Definition 3. The Lipschitz space is the space of functions satisfying where .

Remark 4. When , , where is the homogeneous Besov-Lipschitz space.

Definition 5 (see [14]). Let ; a function is called -atom, where , if it satisfies the following conditions:

As a proper subspace of , the atomic Hardy space is defined by where each is a -atom and is a tempered distribution. Set norm of by where the infimum has taken over all the decompositions of as above.

Given a positive integer and , we denote by the family of all finite subsets of of different elements. For , set . For and , set , , and .

Definition 6. Let () be a locally integrable functions, and . A bounded measurable function on is called -atom, if(i),(ii),(iii) for any , .

A temperate distribution (see [15, 16]) is said to belong to , if, in the Schwartz distribution sense, it can be written as where is -atom, , and . Moreover, , where the infimum has taken over all the decompositions of as above.

Remark 7. When , .
Let and for . Denote .

Definition 8. Let ,   and  .(i)The homogeneous Herz space is defined by where (ii)The nonhomogeneous Herz space is defined by where (the usual modifications are made when ).

Remark 9. When ,  , and .

Definition 10 (see [15]). Let , and .(i)The homogeneous Herz-type Hardy space is defined by and we define .(ii)The nonhomogeneous Herz-type Hardy space is defined by and we define .

Remark 11. When ,   and . And when , we know that , where . However, when ,   and (see [17, 18]).

2. (, ) Bounds of Fractional Hardy Operators

Theorem 12. Let   and  .   maps into .

Proof. Assume that is an atom of and satisfies the following conditions: (i)  , (ii)  , and (iii)  , where , . We now take ; then satisfies: (i)   , (ii)  , and (iii)  .
Suppose that for . Consider where we used the condition . For , we have the following estimate where we used the condition and . For , since , we have . Then The proof is completed.

3. Estimates for Commutators of Fractional Hardy Operators on Hardy Spaces

Definition 13 (see [5, 19]). Let be a locally integrable function on . The commutator of -dimensional Hardy operators is defined by
Meanwhile, the commutators of -dimensional fractional Hardy operators are defined by

In general, the properties of commutator are worse than those of the operators themselves (e.g., the Hardy operators [11] and the singular integral operators [20]). Therefore, when is in , we prove that is not bounded from to . Furthermore, we conclude that the commutator maps from to and the commutator maps into , where and .

Proposition 14. If ,   and , then is not bounded from to .

Proof. We give the proof only for the case , then . Taking and , it is easy to see that . Then for , we have the following estimate: Here is the constant dependent on . So we get

Theorem 15. Let ,  , and then maps into .

Proof. It is enough to prove that holds for any defined in the proof of Theorem 12 and the constant C is independent of . Suppose that for . Then For the first term, implies ; then we have where . For the last term, by , we have So we obtain that Combining all the above estimates, we complete the proof of Theorem 15.

Theorem 16. Let ,   and and ; then maps into .

Proof. Similar to the proof above, suppose that for . Consider For , by the fact that implies , we have that By , we obtain that . Then maps into .

Theorem 17. Let , , and ; , then the following two conditions are equivalent:(i) is bounded from to ;(ii)for all -atom, .

Proof. By Theorem 16, it is clear that (ii) ⇒ (i) is obvious. We only need to prove (i) ⇒ (ii). We assume that for . Let Then Obviously, , and . For , By , we have .
When , ; then

When is in , the commutator has the similar properties.

Proposition 18. If   and  , then is not bounded from to .

Theorem 19. Let   and  ,  , then maps into .

Remark 20. In 2013, Yu and Lu [21] have given the analogous results of Proposition 18 and Theorem 19.

Theorem 21. Let ,   and , ; then is bounded from to .

Theorem 22. Let and , , and ; then the following two conditions are equivalent:(i) maps into ;(ii)for all -atom, .

4. Estimates for Commutators of Fractional Hardy Operators on Herz-Type Hardy Spaces

The boundedness of on the Herz spaces has been obtained as the following, where .

Proposition 23 (see [22]). Let , , , , , and ; then implies that is bounded from to .

Proposition 24 (see [5]). Let , , , , and ; then implies that is bounded from to .

In this section, we discuss the case .

Definition 25. Let and . A function is called central -atom, if it satisfies the following conditions:(i);(ii);(iii), where .

Lemma 26 (see [18]). Let , and .  Consider that if and only if where each is a central -atom with the support and . Moreover, where the infimum has taken over all above decompositions of .

Theorem 27. Let , ,  , , , , then the following two conditions are equivalent:(i) maps into ;(ii)for all central -atom, .

Proof. (i) (ii). Let be a central -atom, with the support . In fact, if , let such that , and for , set ; then is a central -atom with the support .
Similar to the proof of Theorem 16, we have Then Using the fact that maps into When , we have . On the other hand, , so . Combining all the above estimates, we have . For , by , we have that Set ; then
(ii) ⇒ (i). Similar to the previous proof, by , we obtain . Combining and ,   maps into .

When is in , the commutator also has the similar properties.

Theorem 28. Let ,  ,  , , , and ; then the following two conditions are equivalent:(i) maps into ;(ii)for all central -atom, .

5. Estimates for Multilinear Commutators of Fractional Hardy Operators

Definition 29. The multilinear commutator of fractional Hardy operators is defined by where is a -dimensional vector. When , . When , .

The study of multilinear operators is motivated by a mere quest to generalize the theory of linear operators and by their natural appearance in analysis (see [23–25]). In this section, we consider the multilinear commutators of fractional Hardy operators on Hardy spaces.

Theorem 30. Let ,  ,  , and ; then maps to .

Proof. Similar to the proof above, when , we have so it is enough to prove that By , we obtain , ,   and where and ; we complete the proof of Theorem 30.

Theorem 31. Let ,  ,  , and ; then maps into .

Proof. It is similar to the proof of Theorem 15.

When is in , we suppose the multilinear commutator of fractional Hardy operators maps into , but we cannot prove it. However, we get the following result.

Theorem 32. Let and , , , , then maps into .

Proof. Similar to the proof of Theorem 12, it is enough to prove that where is a center -atom supported on a ball . We write where . By Minkowski inequality, we have For , we have ; then For , For , we have ; then . By the condition of for any , , we have that so, Combining all the above estimates, we complete the proof of Theorem 32.

Conflict of Interests

The authors declare that they do not have any commercial or associative interests that represents a conflict of interests in connection with the work submitted.

Acknowledgments

Jiang Zhou is supported by the National Science Foundation of China (Grant nos. 11261055 and 11161044) and the National Natural Science Foundation of Xinjiang (Grant nos. 2011211A005 and BS120104).