A Weighted Variant of Riemann-Liouville Fractional Integrals on ℝ^𝑛

Abstract

We introduce certain type of weighted variant of Riemann-Liouville fractional integral on ℝ^𝑛 and obtain its sharp bounds on the central Morrey and 𝜆-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 𝜆-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 Cesàro.

1. Introduction

Let 0<𝛼<1. The well-known Riemann-Liouville fractional integral 𝐼_𝛼 is defined by 𝐼_𝛼𝑓(𝑥)∶=1Γ(𝛼)^𝑥0𝑓(𝑡)(𝑥−𝑡)^1−𝛼𝑑𝑡,𝑥>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 𝐼_𝛼, the well-known Riesz potential (the solution of Laplace equation) ℐ_𝛼 with 0<𝛼<𝑛 is defined by setting, for all locally integrable functions 𝑓 on ℝ^𝑛, ℐ_𝛼𝑓(𝑥)∶=𝐶_𝑛,𝛼_ℝ𝑛𝑓(𝑡)|𝑥−𝑡|^𝑛−𝛼𝑑𝑡,𝑥∈ℝ^𝑛,(1.2) where 𝐶_𝑛,𝛼∶=𝜋^𝑛/22^𝛼(Γ(𝛼/2))/(Γ((𝑛−𝛼)/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 ℐ_𝛼.

This paper focused on another generalization, the weighted variants of Riemann-Liouville fractional integrals on ℝ^𝑛. 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 𝐼_𝛼 as 𝑥^𝛼𝐼_𝛼𝑓(𝑥)=¹⁰𝑓(𝑡𝑥)1Γ(𝛼)(1−𝑡)^1−𝛼𝑑𝑡,𝑥>0.(1.3) More generally, we use a positive function (weight function) 𝜔(𝑡) to replace 1/(Γ(𝛼)(1−𝑡)^1−𝛼) in (1.3) and generalize the parameter 𝑥 from the positive axle to the Euclidean space ℝ^𝑛 therein. We then derive a weighted generalization of |𝑥|^𝛼𝐼_𝛼 on ℝ^𝑛, which is called the weighted Hardy operator (originally named weighted Hardy-Littlewood avarage) 𝐻_𝜔.

More precise, let 𝜔 be a positive function on [0,1]. The weighted Hardy operator 𝐻_𝜔 is defined by setting, for all complex-valued measurable functions 𝑓 on ℝ^𝑛 and 𝑥∈ℝ^𝑛, 𝐻_𝜔𝑓(𝑥)∶=¹⁰𝑓(𝑡𝑥)𝜔(𝑡)𝑑𝑡.(1.4) Under certain conditions on 𝜔, Carton-Lebrun and Fosset [4] proved that 𝐻_𝜔 maps 𝐿^𝑝(ℝ^𝑛), 1<𝑝<∞, into itself; moreover, the operator 𝐻_𝜔 commutes with the Hilbert transform when 𝑛=1, and with certain Calderón-Zygmund singular integrals including the Riesz transform when 𝑛≥2. Obviously, for 𝑛=1 and 0<𝛼<1, if we take 𝜔(𝑡)∶=1/(Γ(𝛼)(1−𝑡)^1−𝛼), then as mentioned above, for all 𝑥>0, 𝐻_𝜔𝑓(𝑥)=𝑥^−𝛼𝐼_𝛼𝑓(𝑥).(1.5)

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

Theorem A. Let 1<𝑝<∞. Then, 𝐻_𝜔 is bounded on 𝐿^𝑝(ℝ^𝑛) if and only if 𝔸∶=¹⁰𝑡^−𝑛/𝑝𝜔(𝑡)𝑑𝑡<∞.(1.6) Moreover, ‖‖𝐻_𝜔𝑓‖‖_{𝐿𝑝(ℝ𝑛)→𝐿𝑝(ℝ𝑛)}=𝔸.(1.7)

Remark 1.1. Notice that the condition (1.6) implies that 𝜔 is integrable on [0,1] since ∫¹⁰𝜔(𝑡)𝑑𝑡≤∫¹⁰𝑡^−𝑛/𝑝𝜔(𝑡)𝑑𝑡. We naturally assume 𝜔 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<∞, ‖‖𝐼_𝛼‖‖_{𝐿𝑝(𝑑𝑥)→𝐿𝑝(𝑥−𝑝𝛼𝑑𝑥)}=Γ(1−1/𝑝)Γ(1+𝛼−1/𝑝).(1.8) The constant 𝔸 in (1.6) also seems to be of interest as it equals to 𝑝/(𝑝−1) if 𝜔≡1 and 𝑛=1. In this case, 𝐻_𝜔 is precisely reduced to the classical Hardy operator 𝐻 defined by 𝐻𝑓(𝑥)=1𝑥^𝑥0𝑓(𝑡)𝑑𝑡,𝑥>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‖_𝐿^𝑝_(ℝ⁺_)→𝐿^𝑝_(ℝ⁺₎=𝑝𝑝−1(1.10) with 1<𝑝<∞, can be deduced from Theorem A immediately.

Recall that BMO(ℝ^𝑛) is defined to be the space of all 𝑏∈𝐿_loc(ℝ^𝑛) such that ‖𝑏‖_BMO∶=sup_{𝐵⊂ℝ𝑛}1||𝐵||_𝐵||𝑏(𝑥)−𝑏_𝐵||𝑑𝑥<∞,(1.11) where 𝑏_𝐵=(1/|𝐵|)∫_𝐵𝑏 and the supremum is taken over all balls 𝐵 in ℝ^𝑛 with sides parallel to the axes. It is well known that 𝐿^∞(ℝ^𝑛)⊊BMO(ℝ^𝑛), since BMO(ℝ^𝑛) contains unbounded functions such as log|𝑥|. Another interesting result of Xiao in [5] is that the weighted Hardy operator 𝐻_𝜔 is bounded on BMO(ℝ^𝑛), if and only if ¹⁰𝜔(𝑡)𝑑𝑡<∞.(1.12) Moreover, ‖‖𝐻_𝜔‖‖_{BMO(ℝ𝑛)→BMO(ℝ𝑛)}=¹⁰𝜑(𝑡)𝑑𝑡.(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], Krulić 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 𝜆-central BMO spaces. The study of the central Morrey and 𝜆-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 𝐿^𝑞(1<𝑞<∞) spaces. Beurling [14] extended this idea and defined a pair of dual Banach spaces 𝐴^𝑞 and 𝐵^𝑞, where 1/𝑞+1/𝑞′=1. To be precise, 𝐴^𝑞 is a Banach algebra with respect to the convolution, expressed as a union of certain weighted 𝐿^𝑞 spaces. The space 𝐵^𝑞 is expressed as the intersection of the corresponding weighted 𝐿^𝑞 spaces. Later, Feichtinger [15] observed that the space 𝐵^𝑞 can be equivalently described by the set of all locally 𝑞′-integrable functions 𝑓 satisfying that ‖𝑓‖_𝐵^𝑞′=sup_𝑘≥02^{−𝑘𝑛/𝑞′}‖‖𝑓𝜒_𝑘‖‖_𝑞′<∞,(1.14) where 𝜒₀ is the characteristic function of the unit ball {𝑥∈ℝ^𝑛∶|𝑥|≤1}, 𝜒_𝑘 is the characteristic function of the annulus {𝑥∈ℝ^𝑛∶2^𝑘−1<|𝑥|≤2^𝑘}, 𝑘=1,2,3,…, and ‖⋅‖_𝑞 is the norm in 𝐿^𝑞. By duality, the space 𝐴^𝑞, called Beurling algebra now, can be equivalently described by the set of all locally 𝑞-integrable functions 𝑓 satisfying that ‖𝑓‖_𝐴^𝑞=^∞_𝑘=02^{𝑘𝑛/𝑞}‖‖𝑓𝜒_𝑘‖‖_𝑞<∞.(1.15) Based on these, Chen and Lau [16] and García-Cuerva [17] introduced an atomic space 𝐻𝐴^𝑞 associated with the Beurling algebra 𝐴^𝑞 and identified its dual as the space CMO^𝑞, which is defined to be the space of all locally 𝑞-integrable functions 𝑓 satisfying that sup_𝑅≥1||𝐵(01,𝑅)||_𝐵(0,𝑅)||𝑓(𝑥)−𝑓_𝐵(0,𝑅)||^𝑞𝑑𝑥^1/𝑞<∞.(1.16)

By replacing 𝑘∈ℕ∪{0} with 𝑘∈ℤ in (1.3) and (1.6), we obtain the spaces 𝐴̇^𝑞 and 𝐵̇^𝑞, which are the homogeneous version of the spaces 𝐴^𝑞 and 𝐵^𝑞, and the dual space of 𝐴̇^𝑞 is just 𝐵̇^𝑞. Related to these homogeneous spaces, in [18, 19], Lu and Yang introduced the homogeneous counterparts of HA^𝑞 and CMO^𝑞, denoted by HȦ‌^𝑞 and CṀ‌O^𝑞, respectively. These spaces were originally denoted by HK^𝑞 and CBMO_𝑞 in [18, 19]. Recall that the space CṀ‌O^𝑞 is defined to be the space of all locally 𝑞-integrable functions 𝑓 satisfying that sup_𝑅>0||𝐵(01,𝑅)||_𝐵(0,𝑅)||𝑓(𝑥)−𝑓_𝐵(0,𝑅)||^𝑞𝑑𝑥^1/𝑞<∞.(1.17) It was also proved by Lu and Yang that the dual space of HȦ‌^𝑞 is just CṀ‌O^𝑞.

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

Definition 1.2. Let 𝜆∈ℝ and 1<𝑞<∞. The central Morrey space 𝐵̇^𝑞,𝜆(ℝ^𝑛) is defined to be the space of all locally 𝑞-integrable functions 𝑓 satisfying that ‖𝑓‖_̇𝐵^𝑞,𝜆=sup_𝑅>0||𝐵(0,𝑅1)||^1+𝜆𝑞_𝐵(0,𝑅)||𝑓(𝑥)||^𝑞𝑑𝑥^1/𝑞<∞.(1.18)

Definition 1.3. Let 𝜆<1/𝑛 and 1<𝑞<∞. A function 𝑓∈𝐿^𝑞loc(ℝ^𝑛) is said to belong to the 𝜆-central bounded mean oscillation space CṀ‌O^𝑞,𝜆(ℝ^𝑛) if ‖𝑓‖_Ċ‌MO^𝑞,𝜆=sup_𝑅>0||𝐵(0,𝑅1)||^1+𝜆𝑞_𝐵(0,𝑅)||𝑓(𝑥)−𝑓_𝐵(0,𝑅)||^𝑞𝑑𝑥^1/𝑞<∞.(1.19)

We remark that if two functions which differ by a constant are regarded as a function in the space CṀ‌O^𝑞,𝜆, then CṀ‌O^𝑞,𝜆 becomes a Banach space. Apparently, (1.19) is equivalent to the following condition: sup_𝑅>0inf_𝑐∈ℂ||𝐵(0,𝑅1)||^1+𝜆𝑞_𝐵(0,𝑅)||𝑓(𝑥)−𝑐||^𝑞𝑑𝑥^1/𝑞<∞.(1.20)

Remark 1.4. 𝐵̇^𝑞,𝜆 is a Banach space which is continuously included in CṀ‌O^𝑞,𝜆. One can easily check 𝐵̇^𝑞,𝜆(ℝ^𝑛)={0} if 𝜆<−1/𝑞, 𝐵̇^𝑞,0(ℝ^𝑛)=𝐵̇^𝑞(ℝ^𝑛), 𝐵̇^{𝑞,−1/𝑞}(ℝ^𝑛)=𝐿^𝑞(ℝ^𝑛), and 𝐵̇^𝑞,𝜆(ℝ^𝑛)⊋𝐿^𝑞(ℝ^𝑛) if 𝜆>−1/𝑞. Similar to the classical Morrey space, we only consider the case −1/𝑞<𝜆≤0 in this paper.

Remark 1.5. The space CṀ‌O^𝑞,𝜆 when 𝜆=0 is just the space CṀ‌O^𝑞. It is easy to see that BMO⊂CṀ‌O^𝑞 for all 1<𝑞<∞. When 𝜆∈(0,1/𝑛), then the space CṀ‌O^𝑞,𝜆 is just the central version of the Lipschitz space Lip_𝜆(ℝ^𝑛).

Remark 1.6. If 1<𝑞₁<𝑞₂<∞, then by Hölder's inequality, we know that 𝐵̇^𝑞2,𝜆⊂𝐵̇^𝑞1,𝜆 for 𝜆∈ℝ, and CṀ‌O^𝑞2,𝜆⊂CṀ‌O^𝑞1,𝜆 for 𝜆<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 𝜔 which ensure that the corresponding weighted Hardy operator 𝐻_𝜔 is bounded on 𝐵̇^𝑞,𝜆(ℝ^𝑛) and CṀ‌O^𝑞,𝜆(ℝ^𝑛). 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 Cesàro.

2. Sharp Estimates of 𝐻_𝜔

Let us state our main results.

Theorem 2.1. Let 1<𝑞<∞ and −1/𝑞<𝜆≤0. Then 𝐻_𝜔 is a bounded operator on 𝐵̇^𝑞,𝜆(ℝ^𝑛) if and only if 𝔹∶=¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡<∞.(2.1) Moreover, when (2.1) holds, the operator norm of 𝐻_𝜔 on 𝐵̇^𝑞,𝜆(ℝ^𝑛) is given by ‖‖𝐻_𝜔‖‖_{̇𝐵𝑞,𝜆(ℝ𝑛̇𝐵)→𝑞,𝜆(ℝ𝑛)}=𝔹.(2.2)

Proof. Suppose (2.1) holds. For any 𝑅>0, using Minkowski's inequality, we have ||𝐵(0,𝑅1)||^1+𝜆𝑞_𝐵(0,𝑅)||𝐻_𝜔𝑓(𝑥)||^𝑞𝑑𝑥^1/𝑞≤¹⁰||𝐵(0,𝑅1)||^1+𝜆𝑞_𝐵(0,𝑅)=||𝑓(𝑡𝑥)||^𝑞𝑑𝑥^1/𝑞𝜔(𝑡)𝑑𝑡₁₀||𝐵(0,𝑡1𝑅)||^1+𝜆𝑞_{𝐵(0,𝑡𝑅)}≤‖𝑓‖_{̇𝐵𝑞,𝜆(ℝ𝑛)}||𝑓(𝑥)||^𝑞𝑑𝑥^1/𝑞𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(2.3) It implies that ‖‖𝐻_𝜔‖‖_{̇𝐵𝑞,𝜆(ℝ𝑛̇𝐵)→𝑞,𝜆(ℝ𝑛)}≤¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(2.4) Thus 𝐻_𝜔 maps 𝐵̇^𝑞,𝜆(ℝ^𝑛) into itself.
The proof of the converse comes from a standard calculation. If 𝐻_𝜔 is a bounded operator on 𝐵̇^𝑞,𝜆(ℝ^𝑛), take 𝑓₀(𝑥)=|𝑥|^𝑛𝜆,𝑥∈ℝ^𝑛.(2.5) Then ‖‖𝑓₀‖‖_{̇𝐵𝑞,𝜆(ℝ𝑛)}=Ω^𝑛−𝜆1(𝑛𝑞𝜆+𝑛)^1/𝑞,(2.6) where Ω_𝑛=𝜋^𝑛/2/(Γ(1+𝑛/2)) is the volume of the unit ball in ℝ^𝑛.
We have 𝐻_𝜔𝑓₀=𝑓₀¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡,(2.7)‖‖𝐻_𝜔‖‖_{̇𝐵𝑞,𝜆(ℝ𝑛̇𝐵)→𝑞,𝜆(ℝ𝑛)}≥¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(2.8) (2.8) together with (2.4) yields the desired result.

Corollary 2.2. (i) For 0<𝛼<1, 1<𝑞<∞, and −1/𝑞<𝜆≤0, ‖‖𝐼_𝛼‖‖_{̇𝐵𝑞,𝜆̇𝐵(𝑑𝑥)→𝑞,𝜆(𝑥−𝑞𝛼𝑑𝑥)}=Γ(1+𝜆)Γ(1+𝛼+𝜆).(2.9)(ii) For 1<𝑞<∞ and −1/𝑞<𝜆≤0, ‖𝐻‖_̇𝐵^𝑞,𝜆_→̇𝐵^𝑞,𝜆=11+𝜆.(2.10)

Next, we state the corresponding conclusion for the space CṀ‌O^𝑞,𝜆(ℝ^𝑛).

Theorem 2.3. Let 1<𝑞<∞ and 0≤𝜆<1/𝑛. Then 𝐻_𝜔 is a bounded operator on CṀ‌O^𝑞,𝜆(ℝ^𝑛) if and only if (2.1) holds. Moreover, when (2.1) holds, the operator norm of 𝐻_𝜔 on CṀ‌O^𝑞,𝜆(ℝ^𝑛) is given by ‖‖𝐻_𝜔‖‖_{𝐶̇𝑀𝑂𝑞,𝜆(ℝ𝑛̇)→𝐶𝑀𝑂𝑞,𝜆(ℝ𝑛)}=𝔹.(2.11)

Proof. Suppose (2.1) holds. If 𝑓∈CṀ‌O^𝑞,𝜆(ℝ^𝑛), then for any 𝑅>0 and ball 𝐵(0,𝑅), using Fubini's theorem, we see that 𝐻_𝜔𝑓_𝐵(0,𝑅)=¹⁰||𝐵(01,𝑅)||_𝐵(0,𝑅)𝑓(𝑡𝑥)𝑑𝑥𝜔(𝑡)𝑑𝑡=¹⁰𝑓_{𝐵(0,𝑡𝑅)}𝜔(𝑡)𝑑𝑡.(2.12)
Using Minkowski's inequality, we have ||𝐵(0,𝑅1)||^1+𝜆𝑞_𝐵(0,𝑅)|||𝐻_𝜔𝑓(𝑥)−𝐻_𝜔𝑓_𝐵(0,𝑅)|||^𝑞𝑑𝑥^1/𝑞=1||𝐵(0,𝑅)||^1+𝜆𝑞_𝐵(0,𝑅)||||¹⁰𝑓(𝑡𝑥)−𝑓_{𝐵(0,𝑡𝑅)}𝑑𝑡||||^𝑞𝑑𝑥^1/𝑞≤¹⁰||𝐵(0,𝑅1)||^1+𝜆𝑞_𝐵(0,𝑅)=||𝑓(𝑡𝑥)−𝑓_{𝐵(0,𝑡𝑅)}||^𝑞𝑑𝑥^1/𝑞𝜔(𝑡)𝑑𝑡₁₀||𝐵(0,𝑡1𝑅)||^1+𝜆𝑞_{𝐵(0,𝑡𝑅)}≤‖𝑓‖_{Ċ‌MO𝑞,𝜆(ℝ𝑛)}||𝑓(𝑥)−𝑓_{𝐵(0,𝑡𝑅)}||^𝑞𝑑𝑥^1/𝑞𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡,(2.13) which implies 𝐻_𝜔 is bounded on CṀ‌O^𝑞,𝜆(ℝ^𝑛) and ‖‖𝐻_𝜔‖‖_{Ċ‌MO𝑞,𝜆(ℝ𝑛)→Ċ‌MO𝑞,𝜆(ℝ𝑛)}≤𝔹.(2.14)
Conversely, if 𝐻_𝜔 is a bounded operator on CṀ‌O^𝑞,𝜆(ℝ^𝑛), take 𝑓₀(𝑥)=|𝑥|^𝑛𝜆−|𝑥|,_𝑛𝜆𝑥∈ℝ^𝑛𝑟,,𝑥∈ℝ^𝑛𝑙,(2.15) where ℝ^𝑛𝑟 and ℝ^𝑛𝑙 denote the right and the left halves of ℝ^𝑛, separated by the hyperplane 𝑥₁=0, and 𝑥₁ is the first coordinate of 𝑥∈ℝ^𝑛.
Thus, by a standard calculation, we see that (𝑓₀)_𝐵(0,𝑅)=0 and ‖‖𝑓₀‖‖_{Ċ‌MO𝑞,𝜆(ℝ𝑛)}=Ω^𝑛−𝜆1(𝑛𝑞𝜆+𝑛)^1/𝑞𝐻_𝜔,𝑓₀=𝑓₀¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(2.16) From this formula we have ‖‖𝐻_𝜔‖‖_{Ċ‌MO𝑞,𝜆(ℝ𝑛)→Ċ‌MO𝑞,𝜆(ℝ𝑛)}≥𝔹.(2.17)
The proof is complete.

Corollary 2.4. (i) For 1<𝑞<∞ and 0≤𝜆<1, we have ‖𝐻‖_Ċ‌MO^𝑞,𝜆_→Ċ‌MO^𝑞,𝜆=11+𝜆.(2.18)(ii) For 1<𝑞<∞, we have ‖𝐻‖_Ċ‌MO^𝑞_→Ċ‌MO^𝑞=1.

3. A Characterization of Weight Functions via Commutators

A well-known result of Coifman et al. [23] states that the commutator generated by Calderón-Zygmund singular integrals and BMO functions is bounded on 𝐿^𝑝(ℝ^𝑛), 1<𝑝<∞. Recently, we introduced the commutators of weighted Hardy operators and BMO functions introduced in [24]. For any locally integrable function 𝑏 on ℝ^𝑛 and integrable function 𝜔∶[0,1]→[0,∞), the commutator of the weighted Hardy operator 𝐻^𝑏𝜔 is defined by 𝐻^𝑏𝜔𝑓∶=𝑏𝐻_𝜔𝑓−𝐻_𝜔(𝑏𝑓).(3.1)

It is easy to see that when 𝑏∈𝐿^∞(ℝ^𝑛) and 𝜔 satisfies the condition (1.6), then the commutator 𝐻^𝑏𝜔 is bounded on 𝐿^𝑝(ℝ^𝑛), 1<𝑝<∞. An interesting choice of 𝑏 is that it belongs to the class of BMO(ℝ^𝑛). When symbols 𝑏∈BMO(ℝ^𝑛), the condition (1.6) on weight functions 𝜔 can not ensure the boundedness of 𝐻^𝑏𝜔 on 𝐿^𝑝(ℝ^𝑛). Via controlling 𝐻^𝑏𝜔 by the Hardy-Littlewood maximal operators instead of sharp maximal functions, we [24] established a sufficient and necessary (more stronger) condition on weight functions 𝜔 which ensures that 𝐻^𝑏𝜔 is bounded on 𝐿^𝑝(ℝ^𝑛), where 1<𝑝<∞. More recently, Fu and Lu [25] studied the boundedness of 𝐻^𝑏𝜔 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 𝑚-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 𝜆-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<𝑞₁<𝑞<∞,1/𝑞₁=1/𝑞+1/𝑞₂, −1/𝑞<𝜆<0. Assume further that 𝜔 is a positive integrable function on [0,1]. Then, the commutator 𝐻^𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆(ℝ^𝑛), for any 𝑏∈CṀ‌O^𝑞2(ℝ^𝑛), if and only if ℂ∶=¹⁰𝑡^𝑛𝜆𝜔(𝑡)log2𝑡𝑑𝑡<∞.(3.2)

Remark 3.2. The condition (2.1), that is, 𝔹<∞, is weaker than ℂ<∞. In fact, let 𝔻∶=¹⁰𝑡^𝑛𝜆𝜔(𝑡)log1𝑡𝑑𝑡<∞.(3.3)

By ℂ=𝔹log2+𝔻, we know that ℂ<∞ implies 𝔹<∞. But the following example shows that 𝔹<∞ does not imply ℂ<∞. For 0<𝛽<1, if we take 𝑒^{𝑠(−𝑛𝜆−1)}𝜔(𝑠)=⎧⎪⎨⎪⎩𝑠^−1+𝛽𝑠,0<𝑠≤1,_−1−𝛽0,,1𝑠=<0𝑠,<∞∞,,(3.4) and 𝜔(𝑡)=𝜔(log(1/𝑡)), where 0≤𝑡≤1, then 𝔹<∞ and ℂ=∞.

Proof. (i) Let 𝑅∈(0,∞). Denote 𝐵(0,𝑅) by 𝐵 and 𝐵(0,𝑡𝑅) by 𝑡𝐵. Assume ℂ<∞. We get ||𝐵1||_𝐵||𝐻^𝑏𝜔𝑓(𝑥)||^𝑞1𝑑𝑥^1/𝑞1≤1||𝐵||_𝐵¹⁰||(𝑏(𝑥)−𝑏(𝑡𝑥))𝑓(𝑡𝑥)||𝜔(𝑡)𝑑𝑡^𝑞1𝑑𝑥^1/𝑞1≤1||𝐵||_𝐵¹⁰||𝑏(𝑥)−𝑏_𝐵𝑓(𝑡𝑥)||𝜔(𝑡)𝑑𝑡^𝑞1𝑑𝑥^1/𝑞1+1||𝐵||_𝐵¹⁰||𝑏_𝐵−𝑏_𝑡𝐵𝑓(𝑡𝑥)||𝜔(𝑡)𝑑𝑡^𝑞1𝑑𝑥^1/𝑞1+1||𝐵||_𝐵¹⁰||𝑏(𝑡𝑥)−𝑏_𝑡𝐵𝑓(𝑡𝑥)||𝜔(𝑡)𝑑𝑡^𝑞1𝑑𝑥^1/𝑞1∶=𝐼₁+𝐼₂+𝐼₃.(3.5)
By the Minkowski inequality and the Hölder inequality (with 1/𝑞₁=1/𝑞+1/𝑞₂), we have 𝐼₁≤¹⁰||𝐵1||_𝐵≤||𝑏(𝑥)−𝑏_𝐵𝑓(𝑡𝑥)||^𝑞1𝑑𝑥^1/𝑞1𝜔(𝑡)𝑑𝑡₁₀||𝐵1||_𝐵||𝑏(𝑥)−𝑏_𝐵||^𝑞2𝑑𝑥^1/𝑞2||𝐵1||_𝐵≤||𝐵||||𝑓(𝑡𝑥)||^𝑞𝑑𝑥^1/𝑞𝜔(𝑡)𝑑𝑡_𝜆‖𝑏‖_Ċ‌MO^𝑞2¹⁰||𝑡𝐵||1^1+𝑞𝜆_𝑡𝐵≤||𝐵||||𝑓(𝑥)||^𝑞𝑑𝑥^1/𝑞𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡_𝜆‖𝑏‖_Ċ‌MO^𝑞2‖𝑓‖_̇𝐵^𝑞,𝜆¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(3.6) Similarly, we have 𝐼₃≤¹⁰||𝐵1||_𝐵≤||𝑏(𝑡𝑥)−𝑏_𝑡𝐵𝑓(𝑡𝑥)||^𝑞1𝑑𝑥^1/𝑞1𝜔(𝑡)𝑑𝑡₁₀||𝑡1𝐵||_𝑡𝐵||𝑏(𝑥)−𝑏_𝑡𝐵||^𝑞2𝑑𝑥^1/𝑞2||𝑡1𝐵||_𝑡𝐵≤||𝐵||||𝑓(𝑥)||^𝑞𝑑𝑥^1/𝑞𝜔(𝑡)𝑑𝑡_𝜆‖𝑏‖_Ċ‌MO^𝑞2¹⁰||𝑡𝐵||1^1+𝑞𝜆_𝑡𝐵≤𝐶||𝐵||_𝜆||𝑓(𝑥)||^𝑞𝑑𝑥^1/𝑞𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡‖𝑏‖_Ċ‌MO^𝑞2‖𝑓‖_̇𝐵^𝑞,𝜆¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(3.7)
Now we estimate 𝐼₂, 𝐼₂≤¹⁰||𝐵1||_𝐵≤‖𝑓‖_{̇𝐵𝑞,𝜆}||𝑓(𝑡𝑥)||^𝑞1𝑑𝑥^1/𝑞1||𝑏_𝐵−𝑏_𝑡𝐵||𝜔(𝑡)𝑑𝑡¹⁰=‖𝑓‖_{̇𝐵𝑞,𝜆∞}||𝑡𝐵||^𝜆||𝑏_𝐵−𝑏_𝑡𝐵||𝜔(𝑡)𝑑𝑡_𝑘=0^2−𝑘_2−𝑘−1≤‖𝑓‖_{̇𝐵𝑞,𝜆∞}||𝑡𝐵||^𝜆||𝑏_𝐵−𝑏_𝑡𝐵||𝜔(𝑡)𝑑𝑡_𝑘=0^2−𝑘_2−𝑘−1||𝑡𝐵||^𝜆^𝑘_𝑖=0||𝑏_2−𝑖𝐵−𝑏₂^−𝑖−1_𝐵||+||𝑏_{2−𝑘−1𝐵}−𝑏_𝑡𝐵||𝜔(𝑡)𝑑𝑡.(3.8) We see that ^𝑘_𝑖=0||𝑏_2−𝑖𝐵−𝑏₂^−𝑖−1_𝐵||≤𝐶^𝑘_𝑖=0||21^−𝑖𝐵||₂^−𝑖_𝐵||𝑏(𝑦)−𝑏_2−𝑖𝐵||^𝑞2𝑑𝑦^1/𝑞2≤𝐶‖𝑏‖_Ċ‌MO^𝑞2(𝑘+1).(3.9) Therefore, 𝐼₂≤𝐶||𝐵||^𝜆‖𝑏‖_Ċ‌MO^𝑞2‖𝑓‖_̇𝐵^𝑞,𝜆¹⁰𝑡^𝑛𝜆𝜔(𝑡)log1𝑡𝑑𝑡.(3.10)
Combining the estimates of 𝐼₁, 𝐼₂, and 𝐼₃, we conclude that 𝐻^𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1(ℝ^𝑛).
Conversely, assume that for any 𝑏∈CṀ‌O^𝑞2, 𝐻^𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞2,𝜆(ℝ^𝑛). We need to show that ℂ<∞. Since ℂ=𝔹log2+𝔻, we will prove that 𝔹<∞ and 𝔻<∞, respectively. To this end, let 𝑏₀(𝑥)=log|𝑥|(3.11) for all 𝑥∈ℝ^𝑛. Then it follows from Remark 1.5 that 𝑏₀∈BMO⊂CṀ‌O^𝑞2, and ‖‖𝐻^𝑏0_𝜔‖‖_{̇𝐵𝑞,𝜆→̇𝐵𝑞1,𝜆}<∞.(3.12)
Let 𝑓₀(𝑥)=|𝑥|^𝑛𝜆,𝑥∈ℝ^𝑛. Then ‖‖𝑓₀‖‖_{̇𝐵𝑞,𝜆}=Ω^𝑛−𝜆1(𝑛𝑞𝜆+𝑛)^1/𝑞𝐻_𝑏0𝜔,𝑓₀(𝑥)=|𝑥|^𝑛𝜆¹⁰𝑡^𝑛𝜆𝜔(𝑡)log1𝑡𝑑𝑡.(3.13)
For 𝜆>−1/𝑞>−1/𝑞₁, we obtain ‖‖𝐻^𝑏0_𝜔𝑓₀‖‖_{̇𝐵𝑞1,𝜆}=Ω^𝑛−𝜆1𝑛𝑞₁𝜆+𝑛^1/𝑞1¹⁰𝑡^𝑛𝜆𝜔(𝑡)log1𝑡𝑑𝑡.(3.14) So, ‖‖𝐻^𝑏0_𝜔‖‖_{̇𝐵𝑞1,𝜆→̇𝐵𝑞,𝜆}≥𝐶_{𝑛,𝜆,𝑞,𝑞1}¹⁰𝑡^𝑛𝜆𝜔(𝑡)log1𝑡𝑑𝑡.(3.15) Therefore, we have 𝔻<∞.(3.16)
On the other hand, ^01/2𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡≤𝐶^01/2𝑡^𝑛𝜆𝜔(𝑡)log1𝑡𝑑𝑡<∞,_11/2𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡<∞,(3.17) since 𝑡^𝑛𝜆 and 𝜔(𝑡) 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 𝜔 is integrable on [0,1]. However, by Remark 1.1, this assumption is reasonable in some sense.

When 𝑏∈CṀ‌O^𝑞2,𝜆2(ℝ^𝑛) with 𝜆₂>0, namely, 𝑏 is a central 𝜆-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<𝑞₁<𝑞<∞, 1/𝑞₁=1/𝑞+1/𝑞₂, −1/𝑞<𝜆<0, −1/𝑞₁<𝜆₁<0, 0<𝜆₂<1/𝑛, and 𝜆₁=𝜆+𝜆₂. If (2.1) holds true, then for all 𝑏∈CṀ‌O^𝑞2,𝜆2(ℝ^𝑛), the corresponding commutator 𝐻^𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆1(ℝ^𝑛).

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≤||𝐵||^𝜆1‖𝑏‖_Ċ‌MO^𝑞22,𝜆‖𝑓‖_̇𝐵^𝑞,𝜆¹⁰𝐼𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡,₃≤||𝐵||^𝜆1‖𝑏‖_Ċ‌MO^𝑞22,𝜆‖𝑓‖_̇𝐵^𝑞,𝜆¹⁰≤||𝐵||𝑡^𝑛𝜆1𝜔(𝑡)𝑑𝑡_𝜆1‖𝑏‖_Ċ‌MO^𝑞22,𝜆‖𝑓‖_̇𝐵^𝑞,𝜆¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(3.19)
For 𝐼₂, we also have 𝐼₂≤‖𝑓‖_̇𝐵^𝑞,𝜆∞_𝑘=0^2−𝑘_2−𝑘−1||𝑡𝐵||^𝜆^𝑘_𝑖=0||𝑏_2−𝑖𝐵−𝑏₂^−𝑖−1_𝐵||+||𝑏_{2−𝑘−1𝐵}−𝑏_𝑡𝐵||𝜔(𝑡)𝑑𝑡.(3.20) Since now 0<𝜆₂<1/𝑛, we see that ^𝑘_𝑖=0||𝑏_2−𝑖𝐵−𝑏₂^−𝑖−1_𝐵||≤𝐶^𝑘_𝑖=0||21^−𝑖𝐵||₂^−𝑖_𝐵||𝑏(𝑦)−𝑏_2−𝑖𝐵||^𝑞2𝑑𝑦^1/𝑞2≤𝐶‖𝑏‖_Ċ‌MO^𝑞22,𝜆||𝐵||^𝜆2𝑘_𝑖=02^{−𝑖𝑛𝜆2}≤𝐶‖𝑏‖_Ċ‌MO^𝑞22,𝜆||𝐵||^𝜆2.(3.21) Therefore, 𝐼₂≤𝐶||𝐵||^𝜆1‖𝑏‖_Ċ‌MO^𝑞22.𝜆‖𝑓‖_̇𝐵^𝑞,𝜆¹⁰𝑡^𝑛𝜆𝜔(𝑡)𝑑𝑡.(3.22)
Combining the estimates of 𝐼₁, 𝐼₂, and 𝐼₃, we conclude that 𝐻^𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆1(ℝ^𝑛).

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 𝐻^𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆1(ℝ^𝑛) for all 𝑏∈CṀ‌O^𝑞2,𝜆2(ℝ^𝑛) induces (2.1)?

More general, we may extend the previous results to the 𝑘th order commutator of the weighted Hardy operator. Given 𝑘≥1 and a vector ^→𝑏=(𝑏₁,…,𝑏_𝑘), we define the higher order commutator of the weighted Hardy operator as 𝐻^→𝑏𝜔𝑓(𝑥)=¹⁰^𝑘_𝑗=1𝑏_𝑗(𝑥)−𝑏_𝑗(𝑡𝑥)𝑓(𝑡𝑥)𝜔(𝑡)𝑑𝑡,𝑥∈ℝ^𝑛.(3.23) When 𝑘=0, we understand that 𝐻^→𝑏𝜔=𝐻_𝜔. Notice that if 𝑘=1, then 𝐻^→𝑏𝜔=𝐻^𝑏𝜔.

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 𝑘≥2, 1<𝑞₁<𝑞,𝑞₂,…,𝑞_𝑘<∞, 1/𝑞₁=1/𝑞+∑^𝑘𝑖=21/𝑞_𝑖, −1/𝑞<𝜆<0, −1/𝑞₁<𝜆₁<0, 0≤𝜆₂,…,𝜆_𝑘<1/𝑛, and 𝜆₁=𝜆+∑^𝑘𝑖=2𝜆_𝑖.
(i) Assume further that 𝜔 is a positive integrable function on [0,1]. The commutator 𝐻^→𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆(ℝ^𝑛), for any ^→𝑏=(𝑏₂,…,𝑏_𝑘)∈CṀ‌O^𝑞2(ℝ^𝑛)×⋯×CṀ‌O^𝑞𝑘(ℝ^𝑛), if and only if ¹⁰𝑡^𝑛𝜆𝜔(𝑡)log2𝑡^𝑘−1𝑑𝑡<∞.(3.24)(ii) Let 𝜆₂,…,𝜆_𝑘>0 and ^→𝑏=(𝑏₂,…,𝑏_𝑘)∈CṀ‌O^𝑞2,𝜆2(ℝ^𝑛)×⋯×CṀ‌O^{𝑞𝑘,𝜆𝑘}(ℝ^𝑛). If (2.1) holds true, then the corresponding commutator 𝐻^→𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆1(ℝ^𝑛).

Proof. Let 𝑅∈(0,∞). Denote 𝐵(0,𝑅) by 𝐵 and 𝐵(0,𝑡𝑅) by 𝑡𝐵. Assume ℂ<∞. We get ||𝐵1||_𝐵||||𝐻^→_𝑏𝜔𝑓(𝑥)||||^𝑞1𝑑𝑥^1/𝑞1≤⎧⎪⎨⎪⎩1||𝐵||_𝐵⎡⎢⎢⎣¹⁰|||||_𝑘_𝑗=2𝑏_𝑗(𝑥)−𝑏_𝑗(𝑡𝑥)𝑓(𝑡𝑥)|||||𝜔(𝑡)𝑑𝑡⎤⎥⎥⎦^𝑞1𝑑𝑥⎫⎪⎬⎪⎭^1/𝑞1≤𝐶_{𝐼⊂{2,…,𝑘}𝐽⊂{2,…}_{,𝑘},𝐽∩𝐼=∅}⎧⎪⎨⎪⎩1||𝐵||_𝐵⎡⎢⎢⎣¹⁰|||||⎛⎜⎜⎝_𝑖∈𝐼_𝑗∈𝐽_{𝑚∈{2,…,𝑘}⧵(𝐼∪𝐽)}𝑏_𝑖×𝑏(𝑥)−𝑏_𝑖(𝑡𝑥)_𝑗(𝑥)−𝑏_𝑗_𝐵𝑏_𝑚(𝑡𝑥)−𝑏_𝑚_𝑡𝐵𝑓(𝑡𝑥)|||||𝜔(𝑡)𝑑𝑡⎤⎥⎥⎦^𝑞1𝑑𝑥⎫⎪⎬⎪⎭^1/𝑞1.(3.25) Then, applying the Minkowski inequality and the Hölder inequality (with 1/𝑞₁=1/𝑞+∑^𝑘𝑖=21/𝑞_𝑖), and repeating the arguments in the proof of Theorem 3.1, 𝐻^→𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1(ℝ^𝑛) for any ^→𝑏=(𝑏₂,…,𝑏_𝑘)∈CṀ‌O^𝑞2(ℝ^𝑛)×⋯×CṀ‌O^𝑞𝑘(ℝ^𝑛), provided ¹⁰𝑡^𝑛𝜆𝜔(𝑡)log2𝑡^𝑘−1𝑑𝑡<∞.(3.26)
Conversely, assume that 𝐻^→𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1(ℝ^𝑛) for any ^→𝑏=(𝑏₂,…,𝑏_𝑘)∈CṀ‌O^𝑞2(ℝ^𝑛)×⋯×CṀ‌O^𝑞𝑘(ℝ^𝑛). We choose ^→𝑏=(𝑏₂,…,𝑏_𝑘) with 𝑏_𝑗(𝑥)=log|𝑥| for all 𝑥∈ℝ^𝑛 and 𝑗∈{2,…,𝑘}. Then ^→𝑏∈CṀ‌O^𝑞2(ℝ^𝑛)×⋯×CṀ‌O^𝑞𝑘(ℝ^𝑛). 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 Cesàro operator 𝐺_𝜔 is defined by 𝐺_𝜔𝑓(𝑥)=¹⁰𝑓𝑥𝑡𝑡^−𝑛𝜔(𝑡)𝑑𝑡,𝑥∈ℝ^𝑛.(4.1)

If 0<𝛼<1, 𝑛=1, and 𝜔(𝑡)=1/(Γ(𝛼)((1/𝑡)−1)^1−𝛼), then 𝐺_𝜔𝑓(⋅) is reduced to (⋅)^1−𝛼𝐽_𝛼𝑓(⋅), where 𝐽_𝛼 is a variant of Weyl integral operator and defined by 𝐽_𝛼𝑓(𝑥)=1Γ(𝛼)^∞𝑥𝑓(𝑡)(𝑡−𝑥)^1−𝛼𝑑𝑡𝑡(4.2) for all 𝑥∈(0,∞). When 𝜔≡1 and 𝑛=1, 𝐺_𝜔 is the classical Cesàro operator: 𝐺𝑓(𝑥)=⎧⎪⎨⎪⎩∫^∞𝑥𝑓(𝑦)−∫𝑦𝑑𝑦,𝑥>0,_𝑥−∞𝑓(𝑦)𝑦𝑑𝑦,𝑥<0.(4.3)

It was pointed out in [5] that the weighted Hardy operator 𝐻_𝜔 and the weighted Cesàro operator 𝐺_𝜔 are adjoint mutually, namely, _ℝ𝑛𝑔(𝑥)𝐻_𝜔𝑓(𝑥)𝑑𝑥=_ℝ𝑛𝑓(𝑥)𝐺_𝜔𝑔(𝑥)𝑑𝑥(4.4) for all admissible pairs 𝑓 and 𝑔.

Since 𝐴̇^𝑞 and 𝐵̇^𝑞 are a pair of dual Banach spaces, it follows from Theorem 2.1 the following.

Theorem 4.1. Let 1<𝑞<∞. Then 𝐺_𝜔 is bounded on 𝐴̇^𝑞(ℝ^𝑛) if and only if 𝔼∶=¹⁰𝜔(𝑡)𝑑𝑡<∞.(4.5) Moreover, when (4.5) holds, the operator norm of 𝐺_𝜔 on 𝐴̇^𝑞(ℝ^𝑛) is given by ‖‖𝐺_𝜔‖‖_{̇𝐴𝑞(ℝ𝑛̇𝐴)→𝑞(ℝ𝑛)}=𝔼.(4.6)

Corollary 4.2. (i) For 0<𝛼<1 and 1<𝑞<∞, ‖‖𝐽_𝛼‖‖_{̇𝐴𝑞̇𝐴(𝑑𝑥)→𝑞(𝑥𝑞(1−𝛼)𝑑𝑥)}=Γ(1)Γ(1+𝛼).(4.7)(ii) For 1<𝑞<∞, we have ‖𝐺‖_̇𝐴^𝑞_(ℝ^𝑛_̇𝐴)→^𝑞_(ℝ^𝑛₎=1.(4.8)

Since the dual space of 𝐻𝐴̇^𝑞(1<𝑞<∞) is isomorphic to CṀ‌O^𝑞 (see [18, 19]), Theorem 2.3 implies the following result.

Theorem 4.3. Let 1<𝑞<∞. Then 𝐺_𝜔 is a bounded operator on 𝐻𝐴̇^𝑞(ℝ^𝑛) if and only if (4.5) holds. Moreover, when (4.5) holds, the operator norm of 𝐺_𝜔 on 𝐻𝐴̇^𝑞(ℝ^𝑛) is given by ‖‖𝐺_𝜔‖‖_{𝐻̇𝐴𝑞(ℝ𝑛̇𝐴)→𝐻𝑞(ℝ𝑛)}=𝔼.(4.9)

Corollary 4.4. For 1<𝑞<∞, we have ‖𝐺‖_𝐻̇𝐴^𝑞_{̇𝐴→𝐻}^𝑞=1.(4.10)

Following the idea in Section 3, we define the higher order commutator of the weighted Cesàro operator as 𝐺^→𝑏𝜔𝑓(𝑥)=¹⁰^𝑘_𝑗=1𝑏^𝑗𝑥𝑡−𝑏^𝑗(𝑥)𝑓𝑥𝑡𝑡^−𝑛𝜔(𝑡)𝑑𝑡,𝑥∈ℝ^𝑛.(4.11) When 𝑘=0, 𝐺^→𝑏𝜔 is understood as 𝐺_𝜔. Notice that if 𝑘=1, then 𝐺^→𝑏𝜔=𝐺^𝑏𝜔. Similar to the proofs of Theorems 3.1 and 3.3, we have the following result.

Theorem 4.5. Let 𝑘≥2, 1<𝑞₁<𝑞,𝑞₂,…,𝑞_𝑘<∞, 1/𝑞₁=1/𝑞+∑^𝑘𝑖=21/𝑞_𝑖, −1/𝑞<𝜆<0, −1/𝑞₁<𝜆₁<0, 0≤𝜆₂,…,𝜆_𝑘<1/𝑛, and 𝜆₁=𝜆+∑^𝑘𝑖=2𝜆_𝑖.
(i) Assume further that 𝜔 is a positive integrable function on [0,1]. The commutator 𝐺^→𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆(ℝ^𝑛), for any ^→𝑏=(𝑏₂,…,𝑏_𝑘)∈CṀ‌O^𝑞2(ℝ^𝑛)×⋯×CṀ‌O^𝑞𝑘(ℝ^𝑛), if and only if ¹⁰𝑡^{−𝑛(𝜆+1)}𝜔(𝑡)log2𝑡^𝑘−1𝑑𝑡<∞.(4.12)
(ii) Let 𝜆₂,…,𝜆_𝑘>0 and ^→𝑏=(𝑏₂,…,𝑏_𝑘)∈𝐶𝑀̇𝑂^𝑞2,𝜆2(ℝ^𝑛)×⋯×CṀ‌O^{𝑞𝑘,𝜆𝑘}(ℝ^𝑛). Then the corresponding commutator 𝐺^→𝑏𝜔 is bounded from 𝐵̇^𝑞,𝜆(ℝ^𝑛) to 𝐵̇^𝑞1,𝜆1(ℝ^𝑛), provided that ¹⁰𝑡^{−𝑛(𝜆+1)}𝜔(𝑡)𝑑𝑡<∞.(4.13)

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

Let ℕ₀ be the set of all nonnegative integers and 2^−ℕ0 denote the set {2^−𝑗∶𝑗∈ℕ₀}. Let now 𝜑 be a nonnegative function defined on 2^−ℕ0 and 𝑓 be a complex-valued measurable function on ℝ^𝑛. The discrete weighted Hardy operator 𝐻_𝜔 is defined by 𝐻^𝜔𝑓(𝑥)=^∞_𝑘=02^−𝑘𝑓2^−𝑘𝑥𝜔2^−𝑘,𝑥∈ℝ^𝑛,(4.14) and the corresponding discrete weighted Cesàro operator is defined by setting, for all 𝑥∈ℝ^𝑛, 𝐺^𝜔𝑓(𝑥)=^∞_𝑘=0𝑓2^𝑘𝑥2^{𝑘(𝑛−1)}𝜔2^−𝑘.(4.15) We remark that, by the same argument as above with slight modifications, all the results related to the operators 𝐻_𝜔 and 𝐺_𝜔 in Sections 1–4 are also true for their discrete versions 𝐻_𝜔 and 𝐺_𝜔.

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).