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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 780132, 18 pages
http://dx.doi.org/10.1155/2012/780132
Research Article

A Weighted Variant of Riemann-Liouville Fractional Integrals on ℝ𝑛

1Department of Mathematics, Linyi University, Shandong, Linyi 276005, China
2School 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 ℝ𝑛 and obtain its sharp bounds on the central Morrey and πœ†-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 πœ†-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 0<𝛼<1. The well-known Riemann-Liouville fractional integral 𝐼𝛼 is defined by 𝐼𝛼1𝑓(π‘₯)∢=ξ€œΞ“(𝛼)π‘₯0𝑓(𝑑)(π‘₯βˆ’π‘‘)1βˆ’π›Όπ‘‘π‘‘,π‘₯>0,(1.1) for all locally integrable functions 𝑓 on (0,∞). 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 𝑛-dimensional generalization of 𝐼𝛼, the well-known Riesz potential (the solution of Laplace equation) ℐ𝛼 with 0<𝛼<𝑛 is defined by setting, for all locally integrable functions 𝑓 on ℝ𝑛, ℐ𝛼𝑓(π‘₯)∢=𝐢𝑛,π›Όξ€œβ„π‘›π‘“(𝑑)|π‘₯βˆ’π‘‘|π‘›βˆ’π›Όπ‘‘π‘‘,π‘₯βˆˆβ„π‘›,(1.2) where 𝐢𝑛,π›ΌβˆΆ=πœ‹π‘›/22𝛼(Ξ“(𝛼/2))/(Ξ“((π‘›βˆ’π›Ό)/2)). 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 ℐ𝛼.

This paper focused on another generalization, the weighted variants of Riemann-Liouville fractional integrals on ℝ𝑛. 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 𝐼𝛼 as π‘₯π›ΌπΌπ›Όξ€œπ‘“(π‘₯)=101𝑓(𝑑π‘₯)Ξ“(𝛼)(1βˆ’π‘‘)1βˆ’π›Όπ‘‘π‘‘,π‘₯>0.(1.3) More generally, we use a positive function (weight function) πœ”(𝑑) to replace 1/(Ξ“(𝛼)(1βˆ’π‘‘)1βˆ’π›Ό) in (1.3) and generalize the parameter π‘₯ from the positive axle to the Euclidean space ℝ𝑛 therein. We then derive a weighted generalization of |π‘₯|𝛼𝐼𝛼 on ℝ𝑛, which is called the weighted Hardy operator (originally named weighted Hardy-Littlewood avarage) π»πœ”.

More precise, let πœ” be a positive function on [0,1]. The weighted Hardy operator π»πœ” is defined by setting, for all complex-valued measurable functions 𝑓 on ℝ𝑛 and π‘₯βˆˆβ„π‘›, π»πœ”ξ€œπ‘“(π‘₯)∢=10𝑓(𝑑π‘₯)πœ”(𝑑)𝑑𝑑.(1.4) Under certain conditions on πœ”, Carton-Lebrun and Fosset [4] proved that π»πœ” maps 𝐿𝑝(ℝ𝑛), 1<𝑝<∞, into itself; moreover, the operator π»πœ” commutes with the Hilbert transform when 𝑛=1, and with certain CalderΓ³n-Zygmund singular integrals including the Riesz transform when 𝑛β‰₯2. Obviously, for 𝑛=1 and 0<𝛼<1, if we take πœ”(𝑑)∢=1/(Ξ“(𝛼)(1βˆ’π‘‘)1βˆ’π›Ό), then as mentioned above, for all π‘₯>0, π»πœ”π‘“(π‘₯)=π‘₯βˆ’π›ΌπΌπ›Όπ‘“(π‘₯).(1.5)

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

Theorem A. Let 1<𝑝<∞. Then, π»πœ” is bounded on 𝐿𝑝(ℝ𝑛) if and only if ξ€œπ”ΈβˆΆ=10π‘‘βˆ’π‘›/π‘πœ”(𝑑)𝑑𝑑<∞.(1.6) Moreover, β€–β€–π»πœ”π‘“β€–β€–πΏπ‘(ℝ𝑛)→𝐿𝑝(ℝ𝑛)=𝔸.(1.7)

Remark 1.1. Notice that the condition (1.6) implies that πœ” is integrable on [0,1] since ∫10βˆ«πœ”(𝑑)𝑑𝑑≀10π‘‘βˆ’π‘›/π‘πœ”(𝑑)𝑑𝑑. We naturally assume πœ” is integrable on [0,1] throughout this paper.

Obviously, Theorem A implies the celebrated result of Hardy et al. [6, Theorem 329], namely, for all 0<𝛼<1 and 1<𝑝<∞, ‖‖𝐼𝛼‖‖𝐿𝑝(𝑑π‘₯)→𝐿𝑝(π‘₯βˆ’π‘π›Όπ‘‘π‘₯)=Ξ“(1βˆ’1/𝑝).Ξ“(1+π›Όβˆ’1/𝑝)(1.8) The constant 𝔸 in (1.6) also seems to be of interest as it equals to 𝑝/(π‘βˆ’1) if πœ”β‰‘1 and 𝑛=1. In this case, π»πœ” is precisely reduced to the classical Hardy operator 𝐻 defined by 1𝐻𝑓(π‘₯)=π‘₯ξ€œπ‘₯0𝑓(𝑑)𝑑𝑑,π‘₯>0,(1.9) which is the most fundamental integral averaging operator in analysis. Also, a celebrated operator norm estimate due to Hardy et al. [6], that is, ‖𝐻‖𝐿𝑝(ℝ+)→𝐿𝑝(ℝ+)=π‘π‘βˆ’1(1.10) with 1<𝑝<∞, can be deduced from Theorem A immediately.

Recall that BMO(ℝ𝑛) is defined to be the space of all π‘βˆˆπΏloc(ℝ𝑛) such that ‖𝑏‖BMO∢=supπ΅βŠ‚β„π‘›1||𝐡||ξ€œπ΅||𝑏(π‘₯)βˆ’π‘π΅||𝑑π‘₯<∞,(1.11) where π‘π΅βˆ«=(1/|𝐡|)𝐡𝑏 and the supremum is taken over all balls 𝐡 in ℝ𝑛 with sides parallel to the axes. It is well known that 𝐿∞(ℝ𝑛)⊊BMO(ℝ𝑛), since BMO(ℝ𝑛) contains unbounded functions such as log|π‘₯|. Another interesting result of Xiao in [5] is that the weighted Hardy operator π»πœ” is bounded on BMO(ℝ𝑛), if and only if ξ€œ10πœ”(𝑑)𝑑𝑑<∞.(1.12) Moreover, β€–β€–π»πœ”β€–β€–BMO(ℝ𝑛)β†’BMO(ℝ𝑛)=ξ€œ10πœ‘(𝑑)𝑑𝑑.(1.13) 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 πœ†-central BMO spaces. The study of the central Morrey and πœ†-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 πΏπ‘ž(1<π‘ž<∞) spaces. Beurling [14] extended this idea and defined a pair of dual Banach spaces π΄π‘ž and π΅π‘žξ…ž, where 1/π‘ž+1/π‘žβ€²=1. To be precise, π΄π‘ž is a Banach algebra with respect to the convolution, expressed as a union of certain weighted πΏπ‘ž spaces. The space π΅π‘žξ…ž is expressed as the intersection of the corresponding weighted πΏπ‘žξ…ž spaces. Later, Feichtinger [15] observed that the space π΅π‘žξ…ž can be equivalently described by the set of all locally π‘žβ€²-integrable functions 𝑓 satisfying that β€–π‘“β€–π΅π‘žβ€²=supπ‘˜β‰₯0ξ‚€2βˆ’π‘˜π‘›/π‘žβ€²β€–β€–π‘“πœ’π‘˜β€–β€–π‘žβ€²ξ‚<∞,(1.14) where πœ’0 is the characteristic function of the unit ball {π‘₯βˆˆβ„π‘›βˆΆ|π‘₯|≀1}, πœ’π‘˜ is the characteristic function of the annulus {π‘₯βˆˆβ„π‘›βˆΆ2π‘˜βˆ’1<|π‘₯|≀2π‘˜}, π‘˜=1,2,3,…, and β€–β‹…β€–π‘žξ…ž is the norm in πΏπ‘žξ…ž. By duality, the space π΄π‘ž, called Beurling algebra now, can be equivalently described by the set of all locally π‘ž-integrable functions 𝑓 satisfying that β€–π‘“β€–π΄π‘ž=βˆžξ“π‘˜=02π‘˜π‘›/π‘žξ…žβ€–β€–π‘“πœ’π‘˜β€–β€–π‘ž<∞.(1.15) Based on these, Chen and Lau [16] and GarcΓ­a-Cuerva [17] introduced an atomic space π»π΄π‘ž associated with the Beurling algebra π΄π‘ž and identified its dual as the space CMOπ‘ž, which is defined to be the space of all locally π‘ž-integrable functions 𝑓 satisfying that sup𝑅β‰₯1ξ‚΅1||||ξ€œπ΅(0,𝑅)𝐡(0,𝑅)||𝑓(π‘₯)βˆ’π‘“π΅(0,𝑅)||π‘žξ‚Άπ‘‘π‘₯1/π‘ž<∞.(1.16)

By replacing π‘˜βˆˆβ„•βˆͺ{0} with π‘˜βˆˆβ„€ in (1.3) and (1.6), we obtain the spaces Μ‡π΄π‘ž and Μ‡π΅π‘žξ…ž, which are the homogeneous version of the spaces π΄π‘ž and π΅π‘žξ…ž, and the dual space of Μ‡π΄π‘ž is just Μ‡π΅π‘žξ…ž. Related to these homogeneous spaces, in [18, 19], Lu and Yang introduced the homogeneous counterparts of HAπ‘ž and CMOπ‘ž, denoted by Μ‡β€ŒHAπ‘ž and CΜ‡β€ŒMOπ‘ž, respectively. These spaces were originally denoted by HKπ‘ž and CBMOπ‘ž in [18, 19]. Recall that the space CΜ‡β€ŒMOπ‘ž is defined to be the space of all locally π‘ž-integrable functions 𝑓 satisfying that sup𝑅>0ξ‚΅1||||ξ€œπ΅(0,𝑅)𝐡(0,𝑅)||𝑓(π‘₯)βˆ’π‘“π΅(0,𝑅)||π‘žξ‚Άπ‘‘π‘₯1/π‘ž<∞.(1.17) It was also proved by Lu and Yang that the dual space of Μ‡β€ŒHAπ‘ž is just CΜ‡β€ŒMOπ‘ž.

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

Definition 1.2. Let πœ†βˆˆβ„ and 1<π‘ž<∞. The central Morrey space Μ‡π΅π‘ž,πœ†(ℝ𝑛) is defined to be the space of all locally π‘ž-integrable functions 𝑓 satisfying that β€–π‘“β€–Μ‡π΅π‘ž,πœ†=sup𝑅>01||||𝐡(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)||||𝑓(π‘₯)π‘žξƒͺ𝑑π‘₯1/π‘ž<∞.(1.18)

Definition 1.3. Let πœ†<1/𝑛 and 1<π‘ž<∞. A function π‘“βˆˆπΏπ‘žloc(ℝ𝑛) is said to belong to the πœ†-central bounded mean oscillation space CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛) if ‖𝑓‖CΜ‡β€ŒMOπ‘ž,πœ†=sup𝑅>01||||𝐡(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)||𝑓(π‘₯)βˆ’π‘“π΅(0,𝑅)||π‘žξƒͺ𝑑π‘₯1/π‘ž<∞.(1.19)

We remark that if two functions which differ by a constant are regarded as a function in the space CΜ‡β€ŒMOπ‘ž,πœ†, then CΜ‡β€ŒMOπ‘ž,πœ† becomes a Banach space. Apparently, (1.19) is equivalent to the following condition: sup𝑅>0infπ‘βˆˆβ„‚ξƒ©1||||𝐡(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)||||𝑓(π‘₯)βˆ’π‘π‘žξƒͺ𝑑π‘₯1/π‘ž<∞.(1.20)

Remark 1.4. Μ‡π΅π‘ž,πœ† is a Banach space which is continuously included in CΜ‡β€ŒMOπ‘ž,πœ†. One can easily check Μ‡π΅π‘ž,πœ†(ℝ𝑛)={0} if πœ†<βˆ’1/π‘ž, Μ‡π΅π‘ž,0(ℝ𝑛̇𝐡)=π‘ž(ℝ𝑛), Μ‡π΅π‘ž,βˆ’1/π‘ž(ℝ𝑛)=πΏπ‘ž(ℝ𝑛), and Μ‡π΅π‘ž,πœ†(ℝ𝑛)βŠ‹πΏπ‘ž(ℝ𝑛) if πœ†>βˆ’1/π‘ž. Similar to the classical Morrey space, we only consider the case βˆ’1/π‘ž<πœ†β‰€0 in this paper.

Remark 1.5. The space CΜ‡β€ŒMOπ‘ž,πœ† when πœ†=0 is just the space CΜ‡β€ŒMOπ‘ž. It is easy to see that BMOβŠ‚CΜ‡β€ŒMOπ‘ž for all 1<π‘ž<∞. When πœ†βˆˆ(0,1/𝑛), then the space CΜ‡β€ŒMOπ‘ž,πœ† is just the central version of the Lipschitz space Lipπœ†(ℝ𝑛).

Remark 1.6. If 1<π‘ž1<π‘ž2<∞, then by HΓΆlder's inequality, we know that Μ‡π΅π‘ž2,πœ†βŠ‚Μ‡π΅π‘ž1,πœ† for πœ†βˆˆβ„, and CΜ‡β€ŒMOπ‘ž2,πœ†βŠ‚CΜ‡β€ŒMOπ‘ž1,πœ† for πœ†<1/𝑛.

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 πœ” which ensure that the corresponding weighted Hardy operator π»πœ” is bounded on Μ‡π΅π‘ž,πœ†(ℝ𝑛) and CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛). 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 π»πœ”

Let us state our main results.

Theorem 2.1. Let 1<π‘ž<∞ and βˆ’1/π‘ž<πœ†β‰€0. Then π»πœ” is a bounded operator on Μ‡π΅π‘ž,πœ†(ℝ𝑛) if and only if ξ€œπ”ΉβˆΆ=10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑<∞.(2.1) Moreover, when (2.1) holds, the operator norm of π»πœ” on Μ‡π΅π‘ž,πœ†(ℝ𝑛) is given by β€–β€–π»πœ”β€–β€–Μ‡π΅π‘ž,πœ†(ℝ𝑛̇𝐡)β†’π‘ž,πœ†(ℝ𝑛)=𝔹.(2.2)

Proof. Suppose (2.1) holds. For any 𝑅>0, using Minkowski's inequality, we have 1||||𝐡(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)||ξ€·π»πœ”π‘“ξ€Έ||(π‘₯)π‘žξƒͺ𝑑π‘₯1/π‘žβ‰€ξ€œ101||𝐡||(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)||||𝑓(𝑑π‘₯)π‘žξƒͺ𝑑π‘₯1/π‘ž=ξ€œπœ”(𝑑)𝑑𝑑101||||𝐡(0,𝑑𝑅)1+πœ†π‘žξ€œπ΅(0,𝑑𝑅)||||𝑓(π‘₯)π‘žξƒͺ𝑑π‘₯1/π‘žπ‘‘π‘›πœ†πœ”(𝑑)π‘‘π‘‘β‰€β€–π‘“β€–Μ‡π΅π‘ž,πœ†(ℝ𝑛)ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(2.3) It implies that β€–β€–π»πœ”β€–β€–Μ‡π΅π‘ž,πœ†(ℝ𝑛̇𝐡)β†’π‘ž,πœ†(ℝ𝑛)β‰€ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(2.4) Thus π»πœ” maps Μ‡π΅π‘ž,πœ†(ℝ𝑛) into itself.
The proof of the converse comes from a standard calculation. If π»πœ” is a bounded operator on Μ‡π΅π‘ž,πœ†(ℝ𝑛), take 𝑓0(π‘₯)=|π‘₯|π‘›πœ†,π‘₯βˆˆβ„π‘›.(2.5) Then ‖‖𝑓0β€–β€–Μ‡π΅π‘ž,πœ†(ℝ𝑛)=Ξ©π‘›βˆ’πœ†1(π‘›π‘žπœ†+𝑛)1/π‘ž,(2.6) where Ω𝑛=πœ‹π‘›/2/(Ξ“(1+𝑛/2)) is the volume of the unit ball in ℝ𝑛.
We have π»πœ”π‘“0=𝑓0ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑,(2.7)β€–β€–π»πœ”β€–β€–Μ‡π΅π‘ž,πœ†(ℝ𝑛̇𝐡)β†’π‘ž,πœ†(ℝ𝑛)β‰₯ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(2.8) (2.8) together with (2.4) yields the desired result.

Corollary 2.2. (i) For 0<𝛼<1, 1<π‘ž<∞, and βˆ’1/π‘ž<πœ†β‰€0, β€–β€–πΌπ›Όβ€–β€–Μ‡π΅π‘ž,πœ†Μ‡π΅(𝑑π‘₯)β†’π‘ž,πœ†(π‘₯βˆ’π‘žπ›Όπ‘‘π‘₯)=Ξ“(1+πœ†).Ξ“(1+𝛼+πœ†)(2.9)(ii) For 1<π‘ž<∞ and βˆ’1/π‘ž<πœ†β‰€0, β€–π»β€–Μ‡π΅π‘ž,πœ†β†’Μ‡π΅π‘ž,πœ†=1.1+πœ†(2.10)

Next, we state the corresponding conclusion for the space CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛).

Theorem 2.3. Let 1<π‘ž<∞ and 0β‰€πœ†<1/𝑛. Then π»πœ” is a bounded operator on CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛) if and only if (2.1) holds. Moreover, when (2.1) holds, the operator norm of π»πœ” on CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛) is given by β€–β€–π»πœ”β€–β€–πΆΜ‡π‘€π‘‚π‘ž,πœ†(ℝ𝑛̇)β†’πΆπ‘€π‘‚π‘ž,πœ†(ℝ𝑛)=𝔹.(2.11)

Proof. Suppose (2.1) holds. If π‘“βˆˆCΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛), then for any 𝑅>0 and ball 𝐡(0,𝑅), using Fubini's theorem, we see that ξ€·π»πœ”π‘“ξ€Έπ΅(0,𝑅)=ξ€œ10ξ‚΅1||||ξ€œπ΅(0,𝑅)𝐡(0,𝑅)ξ‚Άξ€œπ‘“(𝑑π‘₯)𝑑π‘₯πœ”(𝑑)𝑑𝑑=10𝑓𝐡(0,𝑑𝑅)πœ”(𝑑)𝑑𝑑.(2.12)
Using Minkowski's inequality, we have 1||||𝐡(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)|||ξ€·π»πœ”π‘“ξ€Έξ€·π»(π‘₯)βˆ’πœ”π‘“ξ€Έπ΅(0,𝑅)|||π‘žξƒͺ𝑑π‘₯1/π‘ž=1||𝐡||(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)||||ξ€œ10𝑓(𝑑π‘₯)βˆ’π‘“π΅(0,𝑑𝑅)ξ€Έ||||π‘‘π‘‘π‘žξƒͺ𝑑π‘₯1/π‘žβ‰€ξ€œ101||||𝐡(0,𝑅)1+πœ†π‘žξ€œπ΅(0,𝑅)||𝑓(𝑑π‘₯)βˆ’π‘“π΅(0,𝑑𝑅)||π‘žξƒͺ𝑑π‘₯1/π‘ž=ξ€œπœ”(𝑑)𝑑𝑑101||𝐡||(0,𝑑𝑅)1+πœ†π‘žξ€œπ΅(0,𝑑𝑅)||𝑓(π‘₯)βˆ’π‘“π΅(0,𝑑𝑅)||π‘žξƒͺ𝑑π‘₯1/π‘žπ‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑≀‖𝑓‖CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛)ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑,(2.13) which implies π»πœ” is bounded on CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛) and β€–β€–π»πœ”β€–β€–CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛)β†’CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛)≀𝔹.(2.14)
Conversely, if π»πœ” is a bounded operator on CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛), take 𝑓0ξ‚»(π‘₯)=|π‘₯|π‘›πœ†,π‘₯βˆˆβ„π‘›π‘Ÿ,βˆ’|π‘₯|π‘›πœ†,π‘₯βˆˆβ„π‘›π‘™,(2.15) where β„π‘›π‘Ÿ and ℝ𝑛𝑙 denote the right and the left halves of ℝ𝑛, separated by the hyperplane π‘₯1=0, and π‘₯1 is the first coordinate of π‘₯βˆˆβ„π‘›.
Thus, by a standard calculation, we see that (𝑓0)𝐡(0,𝑅)=0 and ‖‖𝑓0β€–β€–CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛)=Ξ©π‘›βˆ’πœ†1(π‘›π‘žπœ†+𝑛)1/π‘ž,π»πœ”π‘“0=𝑓0ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(2.16) From this formula we have β€–β€–π»πœ”β€–β€–CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛)β†’CΜ‡β€ŒMOπ‘ž,πœ†(ℝ𝑛)β‰₯𝔹.(2.17)
The proof is complete.

Corollary 2.4. (i) For 1<π‘ž<∞ and 0β‰€πœ†<1, we have ‖𝐻‖CΜ‡β€ŒMOπ‘ž,πœ†β†’CΜ‡β€ŒMOπ‘ž,πœ†=1.1+πœ†(2.18)(ii) For 1<π‘ž<∞, we have ‖𝐻‖CΜ‡β€ŒMOπ‘žβ†’CΜ‡β€ŒMOπ‘ž=1.

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 𝐿𝑝(ℝ𝑛), 1<𝑝<∞. Recently, we introduced the commutators of weighted Hardy operators and BMO functions introduced in [24]. For any locally integrable function 𝑏 on ℝ𝑛 and integrable function πœ”βˆΆ[0,1]β†’[0,∞), the commutator of the weighted Hardy operator π»π‘πœ” is defined by π»π‘πœ”π‘“βˆΆ=π‘π»πœ”π‘“βˆ’π»πœ”(𝑏𝑓).(3.1)

It is easy to see that when π‘βˆˆπΏβˆž(ℝ𝑛) and πœ” satisfies the condition (1.6), then the commutator π»π‘πœ” is bounded on 𝐿𝑝(ℝ𝑛), 1<𝑝<∞. An interesting choice of 𝑏 is that it belongs to the class of BMO(ℝ𝑛). When symbols π‘βˆˆBMO(ℝ𝑛), the condition (1.6) on weight functions πœ” can not ensure the boundedness of π»π‘πœ” on 𝐿𝑝(ℝ𝑛). Via controlling π»π‘πœ” by the Hardy-Littlewood maximal operators instead of sharp maximal functions, we [24] established a sufficient and necessary (more stronger) condition on weight functions πœ” which ensures that π»π‘πœ” is bounded on 𝐿𝑝(ℝ𝑛), where 1<𝑝<∞. More recently, Fu and Lu [25] studied the boundedness of π»π‘πœ” 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 π‘š-linear Hardy operators.

Similar to [24], we are devoted to the construction of a sufficient and necessary condition (which is stronger than 𝔹=∞ in Theorem 2.1) on the weight functions so that commutators of weighted Hardy operators (with symbols in πœ†-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 1<π‘ž1<π‘ž<∞,1/π‘ž1=1/π‘ž+1/π‘ž2, βˆ’1/π‘ž<πœ†<0. Assume further that πœ” is a positive integrable function on [0,1]. Then, the commutator π»π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†(ℝ𝑛), for any π‘βˆˆCΜ‡β€ŒMOπ‘ž2(ℝ𝑛), if and only if ξ€œβ„‚βˆΆ=10π‘‘π‘›πœ†2πœ”(𝑑)log𝑑𝑑𝑑<∞.(3.2)

Remark 3.2. The condition (2.1), that is, 𝔹<∞, is weaker than β„‚<∞. In fact, let ξ€œπ”»βˆΆ=10π‘‘π‘›πœ†1πœ”(𝑑)log𝑑𝑑𝑑<∞.(3.3)

By β„‚=𝔹log2+𝔻, we know that β„‚<∞ implies 𝔹<∞. But the following example shows that 𝔹<∞ does not imply β„‚<∞. For 0<𝛽<1, if we take 𝑒𝑠(βˆ’π‘›πœ†βˆ’1)⎧βŽͺ⎨βŽͺβŽ©π‘ ξ‚πœ”(𝑠)=βˆ’1+𝛽𝑠,0<𝑠≀1,βˆ’1βˆ’π›½,1<𝑠<∞,0,𝑠=0,∞,(3.4) and πœ”(𝑑)=ξ‚πœ”(log(1/𝑑)), where 0≀𝑑≀1, then 𝔹<∞ and β„‚=∞.

Proof. (i) Let π‘…βˆˆ(0,∞). Denote 𝐡(0,𝑅) by 𝐡 and 𝐡(0,𝑑𝑅) by 𝑑𝐡. Assume β„‚<∞. We get ξ‚΅1||𝐡||ξ€œπ΅||π»π‘πœ”||𝑓(π‘₯)π‘ž1𝑑π‘₯1/π‘ž1≀1||𝐡||ξ€œπ΅ξ‚΅ξ€œ10||||ξ‚Ά(𝑏(π‘₯)βˆ’π‘(𝑑π‘₯))𝑓(𝑑π‘₯)πœ”(𝑑)π‘‘π‘‘π‘ž1ξƒͺ𝑑π‘₯1/π‘ž1≀1||𝐡||ξ€œπ΅ξ‚΅ξ€œ10||𝑏(π‘₯)βˆ’π‘π΅ξ€Έ||𝑓(𝑑π‘₯)πœ”(𝑑)π‘‘π‘‘π‘ž1ξƒͺ𝑑π‘₯1/π‘ž1+1||𝐡||ξ€œπ΅ξ‚΅ξ€œ10||ξ€·π‘π΅βˆ’π‘π‘‘π΅ξ€Έ||𝑓(𝑑π‘₯)πœ”(𝑑)π‘‘π‘‘π‘ž1ξƒͺ𝑑π‘₯1/π‘ž1+1||𝐡||ξ€œπ΅ξ‚΅ξ€œ10||𝑏(𝑑π‘₯)βˆ’π‘π‘‘π΅ξ€Έ||𝑓(𝑑π‘₯)πœ”(𝑑)π‘‘π‘‘π‘ž1ξƒͺ𝑑π‘₯1/π‘ž1∢=𝐼1+𝐼2+𝐼3.(3.5)
By the Minkowski inequality and the HΓΆlder inequality (with 1/π‘ž1=1/π‘ž+1/π‘ž2), we have 𝐼1β‰€ξ€œ10ξ‚΅1||𝐡||ξ€œπ΅||𝑏(π‘₯)βˆ’π‘π΅ξ€Έ||𝑓(𝑑π‘₯)π‘ž1𝑑π‘₯1/π‘ž1β‰€ξ€œπœ”(𝑑)𝑑𝑑10ξ‚΅1||𝐡||ξ€œπ΅||𝑏(π‘₯)βˆ’π‘π΅||π‘ž2𝑑π‘₯1/π‘ž2ξ‚΅1||𝐡||ξ€œπ΅||𝑓||(𝑑π‘₯)π‘žξ‚Άπ‘‘π‘₯1/π‘žπœ”β‰€||𝐡||(𝑑)π‘‘π‘‘πœ†β€–π‘β€–CΜ‡β€ŒMOπ‘ž2ξ€œ101||||𝑑𝐡1+π‘žπœ†ξ€œπ‘‘π΅||𝑓||(π‘₯)π‘žξƒͺ𝑑π‘₯1/π‘žπ‘‘π‘›πœ†πœ”β‰€||𝐡||(𝑑)π‘‘π‘‘πœ†β€–π‘β€–CΜ‡β€ŒMOπ‘ž2β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(3.6) Similarly, we have 𝐼3β‰€ξ€œ10ξ‚΅1||𝐡||ξ€œπ΅||𝑏(𝑑π‘₯)βˆ’π‘π‘‘π΅ξ€Έ||𝑓(𝑑π‘₯)π‘ž1𝑑π‘₯1/π‘ž1β‰€ξ€œπœ”(𝑑)𝑑𝑑10ξ‚΅1||||ξ€œπ‘‘π΅π‘‘π΅||𝑏(π‘₯)βˆ’π‘π‘‘π΅||π‘ž2𝑑π‘₯1/π‘ž2ξ‚΅1||||ξ€œπ‘‘π΅π‘‘π΅||𝑓||(π‘₯)π‘žξ‚Άπ‘‘π‘₯1/π‘žπœ”β‰€||𝐡||(𝑑)π‘‘π‘‘πœ†β€–π‘β€–CΜ‡β€ŒMOπ‘ž2ξ€œ101||||𝑑𝐡1+π‘žπœ†ξ€œπ‘‘π΅||𝑓||(π‘₯)π‘žξƒͺ𝑑π‘₯1/π‘žπ‘‘π‘›πœ†πœ”||𝐡||(𝑑)π‘‘π‘‘β‰€πΆπœ†β€–π‘β€–CΜ‡β€ŒMOπ‘ž2β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(3.7)
Now we estimate 𝐼2, 𝐼2β‰€ξ€œ10ξ‚΅1||𝐡||ξ€œπ΅||||𝑓(𝑑π‘₯)π‘ž1𝑑π‘₯1/π‘ž1||π‘π΅βˆ’π‘π‘‘π΅||πœ”(𝑑)π‘‘π‘‘β‰€β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10||||π‘‘π΅πœ†||π‘π΅βˆ’π‘π‘‘π΅||πœ”(𝑑)𝑑𝑑=β€–π‘“β€–Μ‡π΅π‘ž,πœ†βˆžξ“π‘˜=0ξ€œ2βˆ’π‘˜2βˆ’π‘˜βˆ’1||||π‘‘π΅πœ†||π‘π΅βˆ’π‘π‘‘π΅||πœ”(𝑑)π‘‘π‘‘β‰€β€–π‘“β€–Μ‡π΅π‘ž,πœ†βˆžξ“π‘˜=0ξ€œ2βˆ’π‘˜2βˆ’π‘˜βˆ’1||||π‘‘π΅πœ†ξƒ―ξƒ©π‘˜ξ“π‘–=0||𝑏2βˆ’π‘–π΅βˆ’π‘2βˆ’π‘–βˆ’1𝐡||ξƒͺ+||𝑏2βˆ’π‘˜βˆ’1π΅βˆ’π‘π‘‘π΅||ξƒ°πœ”(𝑑)𝑑𝑑.(3.8) We see that π‘˜ξ“π‘–=0||𝑏2βˆ’π‘–π΅βˆ’π‘2βˆ’π‘–βˆ’1𝐡||β‰€πΆπ‘˜ξ“π‘–=01||2βˆ’π‘–π΅||ξ€œ2βˆ’π‘–π΅||𝑏(𝑦)βˆ’π‘2βˆ’π‘–π΅||π‘ž2ξƒͺ𝑑𝑦1/π‘ž2≀𝐢‖𝑏‖CΜ‡β€ŒMOπ‘ž2(π‘˜+1).(3.9) Therefore, 𝐼2||𝐡||β‰€πΆπœ†β€–π‘β€–CΜ‡β€ŒMOπ‘ž2β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10π‘‘π‘›πœ†1πœ”(𝑑)log𝑑𝑑𝑑.(3.10)
Combining the estimates of 𝐼1, 𝐼2, and 𝐼3, we conclude that π»π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1(ℝ𝑛).
Conversely, assume that for any π‘βˆˆCΜ‡β€ŒMOπ‘ž2, π»π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž2,πœ†(ℝ𝑛). We need to show that β„‚<∞. Since β„‚=𝔹log2+𝔻, we will prove that 𝔹<∞ and 𝔻<∞, respectively. To this end, let 𝑏0(π‘₯)=log|π‘₯|(3.11) for all π‘₯βˆˆβ„π‘›. Then it follows from Remark 1.5 that 𝑏0∈BMOβŠ‚CΜ‡β€ŒMOπ‘ž2, and ‖‖𝐻𝑏0πœ”β€–β€–Μ‡π΅π‘ž,πœ†β†’Μ‡π΅π‘ž1,πœ†<∞.(3.12)
Let 𝑓0(π‘₯)=|π‘₯|π‘›πœ†,π‘₯βˆˆβ„π‘›. Then ‖‖𝑓0β€–β€–Μ‡π΅π‘ž,πœ†=Ξ©π‘›βˆ’πœ†1(π‘›π‘žπœ†+𝑛)1/π‘ž,𝐻𝑏0πœ”π‘“0(π‘₯)=|π‘₯|π‘›πœ†ξ€œ10π‘‘π‘›πœ†1πœ”(𝑑)log𝑑𝑑𝑑.(3.13)
For πœ†>βˆ’1/π‘ž>βˆ’1/π‘ž1, we obtain ‖‖𝐻𝑏0πœ”π‘“0β€–β€–Μ‡π΅π‘ž1,πœ†=Ξ©π‘›βˆ’πœ†1ξ€·π‘›π‘ž1ξ€Έπœ†+𝑛1/π‘ž1ξ€œ10π‘‘π‘›πœ†1πœ”(𝑑)log𝑑𝑑𝑑.(3.14) So, ‖‖𝐻𝑏0πœ”β€–β€–Μ‡π΅π‘ž1,πœ†β†’Μ‡π΅π‘ž,πœ†β‰₯𝐢𝑛,πœ†,π‘ž,π‘ž1ξ€œ10π‘‘π‘›πœ†1πœ”(𝑑)log𝑑𝑑𝑑.(3.15) Therefore, we have 𝔻<∞.(3.16)
On the other hand, ξ€œ01/2π‘‘π‘›πœ†ξ€œπœ”(𝑑)𝑑𝑑≀𝐢01/2π‘‘π‘›πœ†1πœ”(𝑑)logπ‘‘ξ€œπ‘‘π‘‘<∞,11/2π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑<∞,(3.17) since π‘‘π‘›πœ† and πœ”(𝑑) are integrable functions on [1/2,1]. Combining the above estimates, we get 𝔹<∞.(3.18)
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 πœ” is integrable on [0,1]. However, by Remark 1.1, this assumption is reasonable in some sense.

When π‘βˆˆCΜ‡β€ŒMOπ‘ž2,πœ†2(ℝ𝑛) with πœ†2>0, namely, 𝑏 is a central πœ†-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 1<π‘ž1<π‘ž<∞, 1/π‘ž1=1/π‘ž+1/π‘ž2, βˆ’1/π‘ž<πœ†<0, βˆ’1/π‘ž1<πœ†1<0, 0<πœ†2<1/𝑛, and πœ†1=πœ†+πœ†2. If (2.1) holds true, then for all π‘βˆˆCΜ‡β€ŒMOπ‘ž2,πœ†2(ℝ𝑛), the corresponding commutator π»π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†1(ℝ𝑛).

Proof. Let 𝐼1, 𝐼2, and 𝐼3 be as in the proof of Theorem 3.1. Then, following the estimates of 𝐼1 and 𝐼3 in the proof of Theorem 3.1, we see that 𝐼1≀||𝐡||πœ†1‖𝑏‖CΜ‡β€ŒMOπ‘ž22,πœ†β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10π‘‘π‘›πœ†πΌπœ”(𝑑)𝑑𝑑,3≀||𝐡||πœ†1‖𝑏‖CΜ‡β€ŒMOπ‘ž22,πœ†β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10π‘‘π‘›πœ†1πœ”β‰€||𝐡||(𝑑)π‘‘π‘‘πœ†1‖𝑏‖CΜ‡β€ŒMOπ‘ž22,πœ†β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(3.19)
For 𝐼2, we also have 𝐼2β‰€β€–π‘“β€–Μ‡π΅π‘ž,πœ†βˆžξ“π‘˜=0ξ€œ2βˆ’π‘˜2βˆ’π‘˜βˆ’1||||π‘‘π΅πœ†ξƒ―ξƒ©π‘˜ξ“π‘–=0||𝑏2βˆ’π‘–π΅βˆ’π‘2βˆ’π‘–βˆ’1𝐡||ξƒͺ+||𝑏2βˆ’π‘˜βˆ’1π΅βˆ’π‘π‘‘π΅||ξƒ°πœ”(𝑑)𝑑𝑑.(3.20) Since now 0<πœ†2<1/𝑛, we see that π‘˜ξ“π‘–=0||𝑏2βˆ’π‘–π΅βˆ’π‘2βˆ’π‘–βˆ’1𝐡||β‰€πΆπ‘˜ξ“π‘–=01||2βˆ’π‘–π΅||ξ€œ2βˆ’π‘–π΅||𝑏(𝑦)βˆ’π‘2βˆ’π‘–π΅||π‘ž2ξƒͺ𝑑𝑦1/π‘ž2≀𝐢‖𝑏‖CΜ‡β€ŒMOπ‘ž22,πœ†||𝐡||πœ†2π‘˜ξ“π‘–=02βˆ’π‘–π‘›πœ†2≀𝐢‖𝑏‖CΜ‡β€ŒMOπ‘ž22,πœ†||𝐡||πœ†2.(3.21) Therefore, 𝐼2||𝐡||β‰€πΆπœ†1‖𝑏‖CΜ‡β€ŒMOπ‘ž22.πœ†β€–π‘“β€–Μ‡π΅π‘ž,πœ†ξ€œ10π‘‘π‘›πœ†πœ”(𝑑)𝑑𝑑.(3.22)
Combining the estimates of 𝐼1, 𝐼2, and 𝐼3, we conclude that π»π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†1(ℝ𝑛).

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 π»π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†1(ℝ𝑛) for all π‘βˆˆCΜ‡β€ŒMOπ‘ž2,πœ†2(ℝ𝑛) induces (2.1)?

More general, we may extend the previous results to the π‘˜th order commutator of the weighted Hardy operator. Given π‘˜β‰₯1 and a vector →𝑏=(𝑏1,…,π‘π‘˜), we define the higher order commutator of the weighted Hardy operator as π»β†’π‘πœ”ξ€œπ‘“(π‘₯)=10ξƒ©π‘˜ξ‘π‘—=1𝑏𝑗(π‘₯)βˆ’π‘π‘—ξ€Έξƒͺ(𝑑π‘₯)𝑓(𝑑π‘₯)πœ”(𝑑)𝑑𝑑,π‘₯βˆˆβ„π‘›.(3.23) When π‘˜=0, we understand that π»β†’π‘πœ”=π»πœ”. Notice that if π‘˜=1, then π»β†’π‘πœ”=π»π‘πœ”.

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 π‘˜β‰₯2, 1<π‘ž1<π‘ž,π‘ž2,…,π‘žπ‘˜<∞, 1/π‘ž1βˆ‘=1/π‘ž+π‘˜π‘–=21/π‘žπ‘–, βˆ’1/π‘ž<πœ†<0, βˆ’1/π‘ž1<πœ†1<0, 0β‰€πœ†2,…,πœ†π‘˜<1/𝑛, and πœ†1βˆ‘=πœ†+π‘˜π‘–=2πœ†π‘–.
(i) Assume further that πœ” is a positive integrable function on [0,1]. The commutator π»β†’π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†(ℝ𝑛), for any →𝑏=(𝑏2,…,π‘π‘˜)∈CΜ‡β€ŒMOπ‘ž2(ℝ𝑛)Γ—β‹―Γ—CΜ‡β€ŒMOπ‘žπ‘˜(ℝ𝑛), if and only if ξ€œ10π‘‘π‘›πœ†ξ‚€2πœ”(𝑑)logπ‘‘ξ‚π‘˜βˆ’1𝑑𝑑<∞.(3.24)(ii) Let πœ†2,…,πœ†π‘˜>0 and →𝑏=(𝑏2,…,π‘π‘˜)∈CΜ‡β€ŒMOπ‘ž2,πœ†2(ℝ𝑛)Γ—β‹―Γ—CΜ‡β€ŒMOπ‘žπ‘˜,πœ†π‘˜(ℝ𝑛). If (2.1) holds true, then the corresponding commutator π»β†’π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†1(ℝ𝑛).

Proof. Let π‘…βˆˆ(0,∞). Denote 𝐡(0,𝑅) by 𝐡 and 𝐡(0,𝑑𝑅) by 𝑑𝐡. Assume β„‚<∞. We get 1||𝐡||ξ€œπ΅||||π»β†’π‘πœ”||||𝑓(π‘₯)π‘ž1ξƒͺ𝑑π‘₯1/π‘ž1β‰€βŽ§βŽͺ⎨βŽͺ⎩1||𝐡||ξ€œπ΅βŽ‘βŽ’βŽ’βŽ£ξ€œ10|||||ξƒ©π‘˜ξ‘π‘—=2𝑏𝑗(π‘₯)βˆ’π‘π‘—ξ€Έξƒͺ|||||⎀βŽ₯βŽ₯⎦(𝑑π‘₯)𝑓(𝑑π‘₯)πœ”(𝑑)π‘‘π‘‘π‘ž1⎫βŽͺ⎬βŽͺβŽ­π‘‘π‘₯1/π‘ž1ξ“β‰€πΆπΌβŠ‚{2,…,π‘˜}ξ“π½βŠ‚{2,…,π‘˜},𝐽∩𝐼=βˆ…βŽ§βŽͺ⎨βŽͺ⎩1||𝐡||ξ€œπ΅βŽ‘βŽ’βŽ’βŽ£ξ€œ10|||||βŽ›βŽœβŽœβŽξ‘π‘–βˆˆπΌξ‘π‘—βˆˆπ½ξ‘π‘šβˆˆ{2,…,π‘˜}⧡(𝐼βˆͺ𝐽)𝑏𝑖(π‘₯)βˆ’π‘π‘–ξ€ΈΓ—ξ€·π‘(𝑑π‘₯)𝑗𝑏(π‘₯)βˆ’π‘—ξ€Έπ΅π‘ξ€Έξ€·π‘šξ€·π‘(𝑑π‘₯)βˆ’π‘šξ€Έπ‘‘π΅ξ€Έξƒͺ|||||⎀βŽ₯βŽ₯βŽ¦π‘“(𝑑π‘₯)πœ”(𝑑)π‘‘π‘‘π‘ž1⎫βŽͺ⎬βŽͺβŽ­π‘‘π‘₯1/π‘ž1.(3.25) Then, applying the Minkowski inequality and the HΓΆlder inequality (with 1/π‘ž1βˆ‘=1/π‘ž+π‘˜π‘–=21/π‘žπ‘–), and repeating the arguments in the proof of Theorem 3.1, π»β†’π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1(ℝ𝑛) for any →𝑏=(𝑏2,…,π‘π‘˜)∈CΜ‡β€ŒMOπ‘ž2(ℝ𝑛)Γ—β‹―Γ—CΜ‡β€ŒMOπ‘žπ‘˜(ℝ𝑛), provided ξ€œ10π‘‘π‘›πœ†ξ‚€2πœ”(𝑑)logπ‘‘ξ‚π‘˜βˆ’1𝑑𝑑<∞.(3.26)
Conversely, assume that π»β†’π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1(ℝ𝑛) for any →𝑏=(𝑏2,…,π‘π‘˜)∈CΜ‡β€ŒMOπ‘ž2(ℝ𝑛)Γ—β‹―Γ—CΜ‡β€ŒMOπ‘žπ‘˜(ℝ𝑛). We choose →𝑏=(𝑏2,…,π‘π‘˜) with 𝑏𝑗(π‘₯)=log|π‘₯| for all π‘₯βˆˆβ„π‘› and π‘—βˆˆ{2,…,π‘˜}. Then β†’π‘βˆˆCΜ‡β€ŒMOπ‘ž2(ℝ𝑛)Γ—β‹―Γ—CΜ‡β€ŒMOπ‘žπ‘˜(ℝ𝑛). 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 πΊπœ” is defined by πΊπœ”ξ€œπ‘“(π‘₯)=10𝑓π‘₯π‘‘ξ‚π‘‘βˆ’π‘›πœ”(𝑑)𝑑𝑑,π‘₯βˆˆβ„π‘›.(4.1)

If 0<𝛼<1, 𝑛=1, and πœ”(𝑑)=1/(Ξ“(𝛼)((1/𝑑)βˆ’1)1βˆ’π›Ό), then πΊπœ”π‘“(β‹…) is reduced to (β‹…)1βˆ’π›Όπ½π›Όπ‘“(β‹…), where 𝐽𝛼 is a variant of Weyl integral operator and defined by 𝐽𝛼1𝑓(π‘₯)=ξ€œΞ“(𝛼)∞π‘₯𝑓(𝑑)(π‘‘βˆ’π‘₯)1βˆ’π›Όπ‘‘π‘‘π‘‘(4.2) for all π‘₯∈(0,∞). When πœ”β‰‘1 and 𝑛=1, πΊπœ” is the classical CesΓ ro operator: ⎧βŽͺ⎨βŽͺβŽ©βˆ«πΊπ‘“(π‘₯)=∞π‘₯𝑓(𝑦)π‘¦βˆ’βˆ«π‘‘π‘¦,π‘₯>0,π‘₯βˆ’βˆžπ‘“(𝑦)𝑦𝑑𝑦,π‘₯<0.(4.3)

It was pointed out in [5] that the weighted Hardy operator π»πœ” and the weighted CesΓ ro operator πΊπœ” are adjoint mutually, namely, ξ€œβ„π‘›π‘”(π‘₯)π»πœ”ξ€œπ‘“(π‘₯)𝑑π‘₯=ℝ𝑛𝑓(π‘₯)πΊπœ”π‘”(π‘₯)𝑑π‘₯(4.4) for all admissible pairs 𝑓 and 𝑔.

Since Μ‡π΄π‘ž and Μ‡π΅π‘žξ…ž are a pair of dual Banach spaces, it follows from Theorem 2.1 the following.

Theorem 4.1. Let 1<π‘ž<∞. Then πΊπœ” is bounded on Μ‡π΄π‘ž(ℝ𝑛) if and only if ξ€œπ”ΌβˆΆ=10πœ”(𝑑)𝑑𝑑<∞.(4.5) Moreover, when (4.5) holds, the operator norm of πΊπœ” on Μ‡π΄π‘ž(ℝ𝑛) is given by β€–β€–πΊπœ”β€–β€–Μ‡π΄π‘ž(ℝ𝑛̇𝐴)β†’π‘ž(ℝ𝑛)=𝔼.(4.6)

Corollary 4.2. (i) For 0<𝛼<1 and 1<π‘ž<∞, β€–β€–π½π›Όβ€–β€–Μ‡π΄π‘žΜ‡π΄(𝑑π‘₯)β†’π‘ž(π‘₯π‘ž(1βˆ’π›Ό)𝑑π‘₯)=Ξ“(1).Ξ“(1+𝛼)(4.7)(ii) For 1<π‘ž<∞, we have β€–πΊβ€–Μ‡π΄π‘ž(ℝ𝑛̇𝐴)β†’π‘ž(ℝ𝑛)=1.(4.8)

Since the dual space of π»Μ‡π΄π‘ž(1<π‘ž<∞) is isomorphic to CΜ‡β€ŒMOπ‘žξ…ž (see [18, 19]), Theorem 2.3 implies the following result.

Theorem 4.3. Let 1<π‘ž<∞. Then πΊπœ” is a bounded operator on π»Μ‡π΄π‘ž(ℝ𝑛) if and only if (4.5) holds. Moreover, when (4.5) holds, the operator norm of πΊπœ” on π»Μ‡π΄π‘ž(ℝ𝑛) is given by β€–β€–πΊπœ”β€–β€–π»Μ‡π΄π‘ž(ℝ𝑛̇𝐴)β†’π»π‘ž(ℝ𝑛)=𝔼.(4.9)

Corollary 4.4. For 1<π‘ž<∞, we have β€–πΊβ€–π»Μ‡π΄π‘žΜ‡π΄β†’π»π‘ž=1.(4.10)

Following the idea in Section 3, we define the higher order commutator of the weighted CesΓ ro operator as πΊβ†’π‘πœ”ξ€œπ‘“(π‘₯)=10ξƒ©π‘˜ξ‘π‘—=1𝑏𝑗π‘₯π‘‘ξ‚βˆ’π‘π‘—ξ‚ξƒͺ𝑓π‘₯(π‘₯)π‘‘ξ‚π‘‘βˆ’π‘›πœ”(𝑑)𝑑𝑑,π‘₯βˆˆβ„π‘›.(4.11) When π‘˜=0, πΊβ†’π‘πœ” is understood as πΊπœ”. Notice that if π‘˜=1, then πΊβ†’π‘πœ”=πΊπ‘πœ”. Similar to the proofs of Theorems 3.1 and 3.3, we have the following result.

Theorem 4.5. Let π‘˜β‰₯2, 1<π‘ž1<π‘ž,π‘ž2,…,π‘žπ‘˜<∞, 1/π‘ž1βˆ‘=1/π‘ž+π‘˜π‘–=21/π‘žπ‘–, βˆ’1/π‘ž<πœ†<0, βˆ’1/π‘ž1<πœ†1<0, 0β‰€πœ†2,…,πœ†π‘˜<1/𝑛, and πœ†1βˆ‘=πœ†+π‘˜π‘–=2πœ†π‘–.
(i) Assume further that πœ” is a positive integrable function on [0,1]. The commutator πΊβ†’π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†(ℝ𝑛), for any →𝑏=(𝑏2,…,π‘π‘˜)∈CΜ‡β€ŒMOπ‘ž2(ℝ𝑛)Γ—β‹―Γ—CΜ‡β€ŒMOπ‘žπ‘˜(ℝ𝑛), if and only if ξ€œ10π‘‘βˆ’π‘›(πœ†+1)ξ‚€2πœ”(𝑑)logπ‘‘ξ‚π‘˜βˆ’1𝑑𝑑<∞.(4.12)
(ii) Let πœ†2,…,πœ†π‘˜>0 and →𝑏=(𝑏2,…,π‘π‘˜)βˆˆπΆΜ‡π‘€π‘‚π‘ž2,πœ†2(ℝ𝑛)Γ—β‹―Γ—CΜ‡β€ŒMOπ‘žπ‘˜,πœ†π‘˜(ℝ𝑛). Then the corresponding commutator πΊβ†’π‘πœ” is bounded from Μ‡π΅π‘ž,πœ†(ℝ𝑛) to Μ‡π΅π‘ž1,πœ†1(ℝ𝑛), provided that ξ€œ10π‘‘βˆ’π‘›(πœ†+1)πœ”(𝑑)𝑑𝑑<∞.(4.13)

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

Let β„•0 be the set of all nonnegative integers and 2βˆ’β„•0 denote the set {2βˆ’π‘—βˆΆπ‘—βˆˆβ„•0}. Let now πœ‘ be a nonnegative function defined on 2βˆ’β„•0 and 𝑓 be a complex-valued measurable function on ℝ𝑛. The discrete weighted Hardy operator ξ‚π»πœ” is defined by ξ‚€ξ‚π»πœ”π‘“ξ‚(π‘₯)=βˆžξ“π‘˜=02βˆ’π‘˜π‘“ξ€·2βˆ’π‘˜π‘₯ξ€Έπœ”ξ€·2βˆ’π‘˜ξ€Έ,π‘₯βˆˆβ„π‘›,(4.14) and the corresponding discrete weighted CesΓ ro operator is defined by setting, for all π‘₯βˆˆβ„π‘›, ξ‚€ξ‚πΊπœ”π‘“ξ‚(π‘₯)=βˆžξ“π‘˜=0𝑓2π‘˜π‘₯ξ€Έ2π‘˜(π‘›βˆ’1)πœ”ξ€·2βˆ’π‘˜ξ€Έ.(4.15) We remark that, by the same argument as above with slight modifications, all the results related to the operators π»πœ” and πΊπœ” in Sections 1–4 are also true for their discrete versions ξ‚π»πœ” and ξ‚πΊπœ”.

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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