Weighted Multilinear Hardy Operators on Herz Type Spaces

Abstract

This paper focuses on the bounds of weighted multilinear Hardy operators on the product Herz spaces and the product Morrey-Herz spaces, respectively. We present a sufficient condition on the weight function that guarantees weighted multilinear Hardy operators to be bounded on the product Herz spaces. And the condition is necessary under certain assumptions. Finally, we extend the obtained results to the product Morrey-Herz spaces.

1. Introduction

Let be a nonnegative integral function on . The Hardy operator is defined by . For the operator , Hardy et al. [1] proved that the inequality holds, where and the constant is the best possible. Usually, we call (1) classical Hardy's inequality. With the development of analysis theory, many types of Hardy's inequalities have been discussed. For example, a quite number of papers dealt with the various generalizations, numerous variants, and applications of Hardy's inequalities in the past few years. On the detailed discussions of Hardy's inequalities, we choose to refer to [2, 3].

In 1984, Carton-Lebrun and Fosset [4] gave the definition of the weighted Hardy operator to be where is a measurable function and is a complex-valued measurable function on . It is obvious that degenerates into the classical Hardy operator when , and is defined on . In addition, we call the adjoint operator of the weighted Cesàro average . And the definition of is When and , becomes the classical Cesàro operator It is easy to get that and satisfy when , , . This means that and satisfy the commutative rule .

Under certain conditions on , Carton-Lebrun and Fosset [4] proved that maps into itself for . They also pointed out that the operator commutes with the Hilbert transform when and with certain Calderón-Zygmund singular integral operators including the Riesz transform when . Refer to Xiao [5] for the further extension of the results above; see also [6, 7].

Since Herz space is a natural generalization of weighted Lebesgue spaces with power weights, researchers are also interested in studying the boundedness of on Herz spaces. To make the description more clear below, we review the definition of the Herz spaces now. In the following definitions, , , and , for , , , and , for , and is the characteristic function of a set .

Definition 1 (see [8]). Let , , .
(1) The homogeneous Herz spaces is defined by where
(2) The inhomogenous Herz spaces is defined by where with usual modifications made when or .

In [9], Wu gave the definition of Morrey-Herz spaces.

Definition 2. Let , , , and .
(1) The homogeneous Morrey-Herz space is defined by where
(2) The inhomogeneous Morrey-Herz space is defined by where with usual modifications made when or .

From the above definitions, it is not difficult to note that      for all and . Moreover, the homogenous Herz spaces , the homogenous Morrey-Herz spaces , and the Morrey spaces (see [10, 11]) satisfy . In [12], Liu and Fu discussed the boundedness of the weighted Hardy operator on the Herz space . Conditions on the weighted function were presented to guarantee that is bounded on . They also estimated the corresponding operator norm. And in [13], Fu and Lu further extended the results of [12] to the Morrey-Herz space .

In the past few years, the properties of multilinear operators have also been extensively studied by researchers. There are two reasons for this. First, the multilinear operators are the generalization of the linear ones, and its study makes the research contents of analysis theory more rich. Second, the multilinear operators naturally appear in analysis. The study of multilinear operators is traced to the multilinear singular integral operator theory (see [14]). For more detailed studies on multilinear operators, the readers refer to [15–18] and the references therein. Recently, we have studied the boundedness of weighted multilinear Hardy operators on the product of Lebesgue spaces and central Morrey spaces in [19]. Based on these results and inspired by the results of [12, 13], this paper further concerns the boundedness of on the product Herz spaces and the product Morrey-Herz spaces. We first recall the definition of .

Definition 3. Let , , and be an integrable function. The weighted multilinear Hardy operator is defined by where , and are complex-valued measurable functions on . When , is referred to as bilinear.

In accordance with the case of , we also recall the definition of the weighted multilinear Cesàro operator that is the adjoint operator of .

Definition 4. Let , , and be an integrable function. The weighted multilinear Cesàro operator is defined by where are measurable complex-valued functions on .

Note that and do not satisfy the following commutative rule: This is different from the case of and .

The paper is organized as follows. In Section 2, we present the estimate of the boundedness of on the product Herz spaces. In Section 3, we give the estimates of the boundedness of on the product Morrey-Herz spaces.

2. Boundedness of on the Product of Herz Spaces

We give the first main result of this paper.

Theorem 5. Let and , , , . Then is bounded from to if
Conversely, if , , , , and is bounded from to , then (18) holds. Moreover, in this case, one has

Similarly, we have the following result for the operator .

Theorem 6. Let , , and , , , . If satisfies then is bounded from to .
Conversely, if , , , , and is bounded from to , then (20) holds. Moreover, in this case, one has

Since the proofs of Theorems 5 and 6 are similar, we just give the proof of Theorem 5.

Proof of Theorem 5. In order to simplify the proof, we only consider the case of . Actually, a similar procedure works for all .
Since , by Hölder and Minkowski inequality, we have
For the arbitrary , we can find , such that and . By Minkowski inequality, we have For and , Hölder inequality and Minkowski inequality give
From the above inequality, we have that the first conclusion in Theorem 5 holds.
On the other hand, we suppose that is bounded from to and has the operator norm . For any , let
Obviously, , when . So, for any positive integer , it is easy to get that where . A simple computation gives Similarly, we obtain
When and , we have . So, in this case based on (25). When , we get
Let . It is easy to find a positive integer such that . Thus, we have Since , and , , we have This gives
Using the boundedness of and its operator norm yields Letting in (33) gives Thus, (18) holds.

Since and for all , we deduce the following corollaries from Theorems 5 and 6.

Corollary 7. Let and , , . Then is bounded from to if
Conversely, if , and is bounded from to , then (35) holds. Moreover, in this case, one has

Corollary 8. Let and , , , . If satisfies then is bounded from to .
Conversely, if , , and is bounded from to , then (37) holds. Moreover, in this case, one has

3. Boundedness of on the Product of Morrey-Herz Spaces

Lets give the second main result of this paper.

Theorem 9. Let , and , , , . Then is bounded from to if satisfies
Conversely, when , , , and , , if is bounded from to , then (39) holds. And in this case, one has

For the operator , we have a corresponding result.

Theorem 10. Let , , , , and , , , . Then is bounded from to if satisfies
Conversely, when , , , and , , if is bounded from to , then (41) holds and

Since the proof of Theorem 10 is similar to the proof of Theorem 9, we give the proof of Theorem 9 only.

Proof of Theorem 9. By similarity, we only consider the case that . From the proof of Theorem 5, we know that For and , , Hölder inequality and Minkowski inequality give It means that is bounded from to .
On the other hand, we define When and , we get where . Thus It is similar to obtain For , , and , we have By simple computation, we get Since , , , and , we have Thus, (39) holds in this case.
When and , it is easy to obtain Thus, we have So Then, we get Thus, Since and , we get . Thus This tells us that (39) also holds in this case.
Furthermore, when one of , holds, we suppose but here. From the above computation, we have It gives So, we have Since , , , and , using Taylor expansion yields Thus, we have The above inequality gives Since is bounded from to , we know that (39) holds and The proof of Theorem 9 is complete.

Conflict of Interests

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

Acknowledgments

This work was partially supported by NSF of China (Grant nos. 11271175, 11271162, and 61304023), the AMEP, and DYSP of Linyi University.