# A Weighted Variant of Riemann-Liouville Fractional Integrals onℝn

Authors: Zun Wei Fu; Shan Zhen Lu; Wen Yuan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce certain type of weighted variant of Riemann-Liouville
fractional integral onℝn 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.


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## Body

## 1. Introduction

Let0<α<1. The well-known Riemann-Liouville fractional integralIα is defined by
(1.1)Iαf(x):=1Γ(α)∫0x‍f(t)(x-t)1-αdt,x>0,
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 classicaln-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,
(1.2)ℐαf(x):=Cn,α∫ℝn‍f(t)|x-t|n-αdt,x∈ℝn,
where Cn,α:=πn/22α(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 ℐα.This paper focused on another generalization, the weighted variants of Riemann-Liouville fractional integrals onℝn. 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
(1.3)xαIαf(x)=∫01‍f(tx)1Γ(α)(1-t)1-αdt,x>0.
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 ℝn therein. We then derive a weighted generalization of |x|αIα on ℝn, which is called the weighted Hardy operator (originally named weighted Hardy-Littlewood avarage) Hω.More precise, letω be a positive function on [0,1]. The weighted Hardy operatorHω is defined by setting, for all complex-valued measurable functions f on ℝn and x∈ℝn,
(1.4)Hωf(x):=∫01‍f(tx)ω(t)dt.
Under certain conditions on ω, Carton-Lebrun and Fosset [4] proved that Hω maps Lp(ℝn), 1<p<∞, into itself; moreover, the operator Hω 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,
(1.5)Hωf(x)=x-αIαf(x).A further extension of [4] was due to Xiao [5] as follows.Theorem A.
Let1<p<∞. Then, Hω is bounded on Lp(ℝn) if and only if
(1.6)𝔸:=∫01‍t-n/pω(t)dt<∞.
Moreover,
(1.7)∥Hωf∥Lp(ℝn)→Lp(ℝn)=𝔸.Remark 1.1.
Notice that the condition (1.6) implies that ω is integrable on [0,1] since ∫01‍ω(t)dt≤∫01‍t-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<∞,
(1.8)∥Iα∥Lp(dx)→Lp(x-pαdx)=Γ(1-1/p)Γ(1+α-1/p).
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, Hω is precisely reduced to the classical Hardy operatorH defined by
(1.9)Hf(x)=1x∫0x‍f(t)dt,x>0,
which is the most fundamental integral averaging operator in analysis. Also, a celebrated operator norm estimate due to Hardy et al. [6], that is,
(1.10)∥H∥Lp(ℝ+)→Lp(ℝ+)=pp-1
with 1<p<∞, can be deduced from Theorem A immediately.Recall thatBMO(ℝn) is defined to be the space of all b∈Lloc(ℝn) such that
(1.11)∥b∥BMO:=supB⊂ℝn1|B|∫B‍|b(x)-bB|dx<∞,
where bB=(1/|B|)∫B‍b and the supremum is taken over all balls B 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|x|. Another interesting result of Xiao in [5] is that the weighted Hardy operator Hω is bounded on BMO(ℝn), if and only if
(1.12)∫01‍ω(t)dt<∞.
Moreover,
(1.13)∥Hω∥BMO(ℝn)→BMO(ℝn)=∫01‍φ(t)dt.
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 andλ-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 Lq(1<q<∞) spaces. Beurling [14] extended this idea and defined a pair of dual Banach spaces Aq and Bq', where 1/q+1/q'=1. To be precise, Aq is a Banach algebra with respect to the convolution, expressed as a union of certain weighted Lq spaces. The space Bq' is expressed as the intersection of the corresponding weighted Lq' spaces. Later, Feichtinger [15] observed that the space Bq' can be equivalently described by the set of all locally q'-integrable functions f satisfying that
(1.14)∥f∥Bq'=supk≥0(2-kn/q′∥fχk∥q′)<∞,
where χ0 is the characteristic function of the unit ball {x∈ℝn:|x|≤1}, χk is the characteristic function of the annulus {x∈ℝn:2k-1<|x|≤2k}, k=1,2,3,…, and ∥·∥q' is the norm in Lq'. By duality, the space Aq, called Beurling algebra now, can be equivalently described by the set of all locally q-integrable functions f satisfying that
(1.15)∥f∥Aq=∑k=0∞‍2kn/q'∥fχk∥q<∞.
Based on these, Chen and Lau [16] and García-Cuerva [17] introduced an atomic space HAq associated with the Beurling algebra Aq and identified its dual as the space CMOq, which is defined to be the space of all locally q-integrable functions f satisfying that
(1.16)supR≥1(1|B(0,R)|∫B(0,R)‍|f(x)-fB(0,R)|qdx)1/q<∞.By replacingk∈ℕ∪{0} with k∈ℤ in (1.3) and (1.6), we obtain the spaces A˙q and B˙q', which are the homogeneous version of the spaces Aq and Bq', and the dual space of A˙q is just B˙q'. Related to these homogeneous spaces, in [18, 19], Lu and Yang introduced the homogeneous counterparts of HAq and CMOq, denoted by HA˙q and CM˙Oq, respectively. These spaces were originally denoted by HKq and CBMOq in [18, 19]. Recall that the space CM˙Oq is defined to be the space of all locally q-integrable functions f satisfying that
(1.17)supR>0(1|B(0,R)|∫B(0,R)‍|f(x)-fB(0,R)|qdx)1/q<∞.
It was also proved by Lu and Yang that the dual space of HA˙q is just CM˙Oq.In 2000, Alvarez et al. [20] introduced the following λ-central bounded mean oscillation spaces and the central Morrey spaces, respectively.Definition 1.2.
Letλ∈ℝ and 1<q<∞. The central Morrey space B˙q,λ(ℝn) is defined to be the space of all locally q-integrable functions f satisfying that
(1.18)∥f∥B˙q,λ=supR>0(1|B(0,R)|1+λq∫B(0,R)‍|f(x)|qdx)1/q<∞.Definition 1.3.
Letλ<1/n and 1<q<∞. A function f∈Llocq(ℝn) is said to belong to the λ-central bounded mean oscillation space CM˙Oq,λ(ℝn) if
(1.19)∥f∥CM˙Oq,λ=supR>0(1|B(0,R)|1+λq∫B(0,R)‍|f(x)-fB(0,R)|qdx)1/q<∞.We remark that if two functions which differ by a constant are regarded as a function in the spaceCM˙Oq,λ, then CM˙Oq,λ becomes a Banach space. Apparently, (1.19) is equivalent to the following condition:
(1.20)supR>0infc∈ℂ(1|B(0,R)|1+λq∫B(0,R)‍|f(x)-c|qdx)1/q<∞.Remark 1.4.
B
˙
q
,
λ is a Banach space which is continuously included in CM˙Oq,λ. One can easily check B˙q,λ(ℝn)={0} if λ<-1/q, B˙q,0(ℝn)=B˙q(ℝn), B˙q,-1/q(ℝn)=Lq(ℝn), and B˙q,λ(ℝn)⊋Lq(ℝn) 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 spaceCM˙Oq,λ when λ=0 is just the space CM˙Oq. It is easy to see that BMO⊂CM˙Oq for all 1<q<∞. When λ∈(0,1/n), then the space CM˙Oq,λ is just the central version of the Lipschitz space Lipλ(ℝn).Remark 1.6.
If1<q1<q2<∞, then by Hölder's inequality, we know that B˙q2,λ⊂B˙q1,λ for λ∈ℝ, and CM˙Oq2,λ⊂CM˙Oq1,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 weightω which ensure that the corresponding weighted Hardy operator Hω is bounded on B˙q,λ(ℝn) and CM˙Oq,λ(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 Cesàro.


## 2. Sharp Estimates ofHω

Let us state our main results.Theorem 2.1.
Let1<q<∞ and -1/q<λ≤0. Then Hω is a bounded operator on B˙q,λ(ℝn) if and only if
(2.1)𝔹:=∫01‍tnλω(t)dt<∞.
Moreover, when (2.1) holds, the operator norm of Hω on B˙q,λ(ℝn) is given by
(2.2)∥Hω∥B˙q,λ(ℝn)→B˙q,λ(ℝn)=𝔹.Proof.
Suppose (2.1) holds. For any R>0, using Minkowski's inequality, we have
(2.3)(1|B(0,R)|1+λq∫B(0,R)‍|(Hωf)(x)|qdx)1/q≤∫01‍(1|B(0,R)|1+λq∫B(0,R)‍|f(tx)|qdx)1/qω(t)dt=∫01‍(1|B(0,tR)|1+λq∫B(0,tR)‍|f(x)|qdx)1/qtnλω(t)dt≤∥f∥B˙q,λ(ℝn)∫01‍tnλω(t)dt.
It implies that
(2.4)∥Hω∥B˙q,λ(ℝn)→B˙q,λ(ℝn)≤∫01‍tnλω(t)dt.
Thus Hω maps B˙q,λ(ℝn) into itself.
The proof of the converse comes from a standard calculation. IfHω is a bounded operator on B˙q,λ(ℝn), take
(2.5)f0(x)=|x|nλ,x∈ℝn.
Then
(2.6)∥f0∥B˙q,λ(ℝn)=Ωn-λ1(nqλ+n)1/q,
where Ωn=πn/2/(Γ(1+n/2)) is the volume of the unit ball in ℝn.
We have(2.7)Hωf0=f0∫01‍tnλω(t)dt,(2.8)∥Hω∥B˙q,λ(ℝn)→B˙q,λ(ℝn)≥∫01‍tnλω(t)dt.
(2.8) together with (2.4) yields the desired result.Corollary 2.2.
(i)   For0<α<1, 1<q<∞, and -1/q<λ≤0,
(2.9)∥Iα∥B˙q,λ(dx)→B˙q,λ(x-qαdx)=Γ(1+λ)Γ(1+α+λ).(ii)   For 1<q<∞ and -1/q<λ≤0,
(2.10)∥H∥B˙q,λ→B˙q,λ=11+λ.Next, we state the corresponding conclusion for the spaceCM˙Oq,λ(ℝn).Theorem 2.3.
Let1<q<∞ and 0≤λ<1/n. Then Hω is a bounded operator on CM˙Oq,λ(ℝn) if and only if (2.1) holds. Moreover, when (2.1) holds, the operator norm of Hω on CM˙Oq,λ(ℝn) is given by
(2.11)∥Hω∥CM˙Oq,λ(ℝn)→CM˙Oq,λ(ℝn)=𝔹.Proof.
Suppose (2.1) holds. If f∈CM˙Oq,λ(ℝn), then for any R>0 and ball B(0,R), using Fubini's theorem, we see that
(2.12)(Hωf)B(0,R)=∫01‍(1|B(0,R)|∫B(0,R)‍f(tx)dx)ω(t)dt=∫01‍fB(0,tR)ω(t)dt.
Using Minkowski's inequality, we have(2.13)(1|B(0,R)|1+λq∫B(0,R)‍|(Hωf)(x)-(Hωf)B(0,R)|qdx)1/q=(1|B(0,R)|1+λq∫B(0,R)‍|∫01‍(f(tx)-fB(0,tR))dt|qdx)1/q≤∫01‍(1|B(0,R)|1+λq∫B(0,R)‍|f(tx)-fB(0,tR)|qdx)1/qω(t)dt=∫01‍(1|B(0,tR)|1+λq∫B(0,tR)‍|f(x)-fB(0,tR)|qdx)1/qtnλω(t)dt≤∥f∥CM˙Oq,λ(ℝn)∫01‍tnλω(t)dt,
which implies Hω is bounded on CM˙Oq,λ(ℝn) and
(2.14)∥Hω∥CM˙Oq,λ(ℝn)→CM˙Oq,λ(ℝn)≤𝔹.
Conversely, ifHω is a bounded operator on CM˙Oq,λ(ℝn), take
(2.15)f0(x)={|x|nλ,x∈ℝrn,-|x|nλ,x∈ℝln,
where ℝrn and ℝln denote the right and the left halves of ℝn, separated by the hyperplane x1=0, and x1 is the first coordinate of x∈ℝn.
Thus, by a standard calculation, we see that(f0)B(0,R)=0 and
(2.16)∥f0∥CM˙Oq,λ(ℝn)=Ωn-λ1(nqλ+n)1/q,Hωf0=f0∫01‍tnλω(t)dt.
From this formula we have
(2.17)∥Hω∥CM˙Oq,λ(ℝn)→CM˙Oq,λ(ℝn)≥𝔹.
The proof is complete.Corollary 2.4.
(i)   For1<q<∞ and 0≤λ<1, we have
(2.18)∥H∥CM˙Oq,λ→CM˙Oq,λ=11+λ.(ii)   For 1<q<∞, we have ∥H∥CM˙Oq→CM˙Oq=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 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 ω:[0,1]→[0,∞), the commutator of the weighted Hardy operatorHωb is defined by
(3.1)Hωbf:=bHωf-Hω(bf).It is easy to see that whenb∈L∞(ℝn) and ω satisfies the condition (1.6), then the commutator Hωb is bounded on Lp(ℝn), 1<p<∞. An interesting choice of b is that it belongs to the class of BMO(ℝn). When symbols b∈BMO(ℝn), the condition (1.6) on weight functions ω can not ensure the boundedness of Hωb on Lp(ℝn). Via controlling Hωb 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 Hωb is bounded on Lp(ℝn), where 1<p<∞. More recently, Fu and Lu [25] studied the boundedness of Hωb 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.
Let1<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 Hωb is bounded from B˙q,λ(ℝn) to B˙q1,λ(ℝn), for any b∈CM˙Oq2(ℝn), if and only if
(3.2)ℂ:=∫01‍tnλω(t)log2tdt<∞.Remark 3.2.
The condition (2.1), that is, 𝔹<∞, is weaker than ℂ<∞. In fact, let
(3.3)𝔻:=∫01‍tnλω(t)log1tdt<∞.Byℂ=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
(3.4)es(-nλ-1)ω~(s)={s-1+β,0<s≤1,s-1-β,1<s<∞,0,s=0,∞,
and ω(t)=ω~(log(1/t)), where 0≤t≤1, then 𝔹<∞ and ℂ=∞.Proof.
(i) LetR∈(0,∞). Denote B(0,R) by B and B(0,tR) by tB. Assume ℂ<∞. We get
(3.5)(1|B|∫B‍|Hωbf(x)|q1dx)1/q1≤(1|B|∫B‍(∫01‍|(b(x)-b(tx))f(tx)|ω(t)dt)q1dx)1/q1≤(1|B|∫B‍(∫01‍|(b(x)-bB)f(tx)|ω(t)dt)q1dx)1/q1≤≤+(1|B|∫B‍(∫01‍|(bB-btB)f(tx)|ω(t)dt)q1dx)1/q1≤≤+(1|B|∫B‍(∫01‍|(b(tx)-btB)f(tx)|ω(t)dt)q1dx)1/q1:=I1+I2+I3.
By the Minkowski inequality and the Hölder inequality (with1/q1=1/q+1/q2), we have
(3.6)I1≤∫01‍(1|B|∫B‍|(b(x)-bB)f(tx)|q1dx)1/q1ω(t)dt≤∫01‍(1|B|∫B‍|b(x)-bB|q2dx)1/q2(1|B|∫B‍|f(tx)|qdx)1/qω(t)dt≤|B|λ∥b∥CM˙Oq2∫01‍(1|tB|1+qλ∫tB‍|f(x)|qdx)1/qtnλω(t)dt≤|B|λ∥b∥CM˙Oq2∥f∥B˙q,λ∫01‍tnλω(t)dt.
Similarly, we have
(3.7)I3≤∫01‍(1|B|∫B‍|(b(tx)-btB)f(tx)|q1dx)1/q1ω(t)dt≤∫01‍(1|tB|∫tB‍|b(x)-btB|q2dx)1/q2(1|tB|∫tB‍|f(x)|qdx)1/qω(t)dt≤|B|λ∥b∥CM˙Oq2∫01‍(1|tB|1+qλ∫tB‍|f(x)|qdx)1/qtnλω(t)dt≤C|B|λ∥b∥CM˙Oq2∥f∥B˙q,λ∫01‍tnλω(t)dt.
Now we estimateI2,
(3.8)I2≤∫01‍(1|B|∫B‍|f(tx)|q1dx)1/q1|bB-btB|ω(t)dt≤∥f∥B˙q,λ∫01‍|tB|λ|bB-btB|ω(t)dt=∥f∥B˙q,λ∑k=0∞‍∫2-k-12-k‍|tB|λ|bB-btB|ω(t)dt≤∥f∥B˙q,λ∑k=0∞‍∫2-k-12-k‍|tB|λ{(∑i=0k‍|b2-iB-b2-i-1B|)+|b2-k-1B-btB|}ω(t)dt.
We see that
(3.9)∑i=0k‍|b2-iB-b2-i-1B|≤C∑i=0k‍(1|2-iB|∫2-iB‍|b(y)-b2-iB|q2dy)1/q2≤C∥b∥CM˙Oq2(k+1).
Therefore,
(3.10)I2≤C|B|λ∥b∥CM˙Oq2∥f∥B˙q,λ∫01‍tnλω(t)log1tdt.
Combining the estimates ofI1, I2, and I3, we conclude that Hωb is bounded from B˙q,λ(ℝn) to B˙q1(ℝn).
Conversely, assume that for anyb∈CM˙Oq2, Hωb is bounded from B˙q,λ(ℝn) to B˙q2,λ(ℝn). We need to show that ℂ<∞. Since ℂ=𝔹log2+𝔻, we will prove that 𝔹<∞ and 𝔻<∞, respectively. To this end, let
(3.11)b0(x)=log|x|
for all x∈ℝn. Then it follows from Remark 1.5 that b0∈BMO⊂CM˙Oq2, and
(3.12)∥Hωb0∥B˙q,λ→B˙q1,λ<∞.
Letf0(x)=|x|nλ,x∈ℝn. Then
(3.13)∥f0∥B˙q,λ=Ωn-λ1(nqλ+n)1/q,Hωb0f0(x)=|x|nλ∫01‍tnλω(t)log1tdt.
Forλ>-1/q>-1/q1, we obtain
(3.14)∥Hωb0f0∥B˙q1,λ=Ωn-λ1(nq1λ+n)1/q1∫01‍tnλω(t)log1tdt.
So,
(3.15)∥Hωb0∥B˙q1,λ→B˙q,λ≥Cn,λ,q,q1∫01‍tnλω(t)log1tdt.
Therefore, we have
(3.16)𝔻<∞.
On the other hand,(3.17)∫01/2‍tnλω(t)dt≤C∫01/2‍tnλω(t)log1tdt<∞,∫1/21‍tnλω(t)dt<∞,
since tnλ and ω(t) are integrable functions on [1/2,1]. Combining the above estimates, we get
(3.18)𝔹<∞.
Combining (3.18) and (3.16), we then obtain the desired result.Notice that comparing with Theorems2.1 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.Whenb∈CM˙Oq2,λ2(ℝn) 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.
Let1<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∈CM˙Oq2,λ2(ℝn), the corresponding commutator Hωb is bounded from B˙q,λ(ℝn) to B˙q1,λ1(ℝn).Proof.
LetI1, 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
(3.19)I1≤|B|λ1∥b∥CM˙Oq2,λ2∥f∥B˙q,λ∫01‍tnλω(t)dt,I3≤|B|λ1∥b∥CM˙Oq2,λ2∥f∥B˙q,λ∫01‍tnλ1ω(t)dt≤|B|λ1∥b∥CM˙Oq2,λ2∥f∥B˙q,λ∫01‍tnλω(t)dt.
ForI2, we also have
(3.20)I2≤∥f∥B˙q,λ∑k=0∞‍∫2-k-12-k‍|tB|λ{(∑i=0k‍|b2-iB-b2-i-1B|)+|b2-k-1B-btB|}ω(t)dt.
Since now 0<λ2<1/n, we see that
(3.21)∑i=0k‍|b2-iB-b2-i-1B|≤C∑i=0k‍(1|2-iB|∫2-iB‍|b(y)-b2-iB|q2dy)1/q2≤C∥b∥CM˙Oq2,λ2|B|λ2∑i=0k‍2-inλ2≤C∥b∥CM˙Oq2,λ2|B|λ2.
Therefore,
(3.22)I2≤C|B|λ1∥b∥CM˙Oq2.λ2∥f∥B˙q,λ∫01‍tnλω(t)dt.
Combining the estimates ofI1, I2, and I3, we conclude that Hωb is bounded from B˙q,λ(ℝn) to B˙q1,λ1(ℝn).Different from Theorem3.1, it is still unknown whether the condition (2.1) in Theorem 3.3 is sharp. That is, whether the fact that Hωb is bounded from B˙q,λ(ℝn) to B˙q1,λ1(ℝn) for all b∈CM˙Oq2,λ2(ℝn) induces (2.1)?More general, we may extend the previous results to thekth  order commutator of the weighted Hardy operator. Given k≥1 and a vector b→=(b1,…,bk), we define the higher order commutator of the weighted Hardy operator as
(3.23)Hωb→f(x)=∫01‍(∏j=1k‍(bj(x)-bj(tx)))f(tx)ω(t)dt,x∈ℝn.
When k=0, we understand that Hωb→=Hω. Notice that if k=1, then Hωb→=Hωb.Using the method in the proof of Theorems3.1 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.
Letk≥2, 1<q1<q,q2,…,qk<∞, 1/q1=1/q+∑i=2k‍1/qi, -1/q<λ<0, -1/q1<λ1<0, 0≤λ2,…,λk<1/n, and λ1=λ+∑i=2k‍λi.
(i) Assume further thatω is a positive integrable function on [0,1]. The commutator Hωb→ is bounded from B˙q,λ(ℝn) to B˙q1,λ(ℝn), for any b→=(b2,…,bk)∈CM˙Oq2(ℝn)×⋯×CM˙Oqk(ℝn), if and only if
(3.24)∫01‍tnλω(t)(log2t)k-1dt<∞.(ii) Let λ2,…,λk>0 and b→=(b2,…,bk)∈CM˙Oq2,λ2(ℝn)×⋯×CM˙Oqk,λk(ℝn). If (2.1) holds true, then the corresponding commutator Hωb→ is bounded from B˙q,λ(ℝn) to B˙q1,λ1(ℝn).Proof.
LetR∈(0,∞). Denote B(0,R) by B and B(0,tR) by tB. Assume ℂ<∞. We get
(3.25)(1|B|∫B‍|Hωb→f(x)|q1dx)1/q1≤{1|B|∫B‍[∫01‍|(∏j=2k‍(bj(x)-bj(tx)))f(tx)|ω(t)dt]q1dx}1/q1≤C∑I⊂{2,…,k}‍∑J⊂{2,…,k},J∩I=∅‍{1|B|∫B‍[∫01‍|(∏i∈I‍∏j∈J‍∏m∈{2,…,k}∖(I∪J)‍(bi(x)-bi(tx))C∑I⊂{2,⋯,k}‍∑J⊂{2,⋯,k},J∩I=∅++×(bj(x)-(bj)B)(bm(tx)-(bm)tB)∏j∈J‍)f(tx)|ω(t)dt]q1dx}1/q1.
Then, applying the Minkowski inequality and the Hölder inequality (with 1/q1=1/q+∑i=2k‍1/qi), and repeating the arguments in the proof of Theorem 3.1, Hωb→ is bounded from B˙q,λ(ℝn) to B˙q1(ℝn) for any b→=(b2,…,bk)∈CM˙Oq2(ℝn)×⋯×CM˙Oqk(ℝn), provided
(3.26)∫01‍tnλω(t)(log2t)k-1dt<∞.
Conversely, assume thatHωb→ is bounded from B˙q,λ(ℝn) to B˙q1(ℝn) for any b→=(b2,…,bk)∈CM˙Oq2(ℝn)×⋯×CM˙Oqk(ℝn). We choose b→=(b2,…,bk) with bj(x)=log|x| for all x∈ℝn and j∈{2,…,k}. Then b→∈CM˙Oq2(ℝn)×⋯×CM˙Oqk(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 theweighted Cesàro operatorGω is defined by
(4.1)Gωf(x)=∫01‍f(xt)t-nω(t)dt,x∈ℝn.If0<α<1, n=1, and ω(t)=1/(Γ(α)((1/t)-1)1-α), then Gωf(·) is reduced to (·)1-αJαf(·), where Jα is a variant of Weyl integral operator and defined by
(4.2)Jαf(x)=1Γ(α)∫x∞‍f(t)(t-x)1-αdtt
for all x∈(0,∞). When ω≡1 and n=1, Gω is the classical Cesàro operator:
(4.3)Gf(x)={∫x∞f(y)ydy,‍x>0,-∫-∞x‍f(y)ydy,x<0.It was pointed out in [5] that the weighted Hardy operator Hω and the weighted Cesàro operator Gω are adjoint mutually, namely,
(4.4)∫ℝn‍g(x)Hωf(x)dx=∫ℝn‍f(x)Gωg(x)dx
for all admissible pairs f and g.SinceA˙q and B˙q' are a pair of dual Banach spaces, it follows from Theorem 2.1 the following.Theorem 4.1.
Let1<q<∞. Then Gω is bounded on A˙q(ℝn) if and only if
(4.5)𝔼:=∫01‍ω(t)dt<∞.
Moreover, when (4.5) holds, the operator norm of Gω on A˙q(ℝn) is given by
(4.6)∥Gω∥A˙q(ℝn)→A˙q(ℝn)=𝔼.Corollary 4.2.
(i) For0<α<1 and 1<q<∞,
(4.7)∥Jα∥A˙q(dx)→A˙q(xq(1-α)dx)=Γ(1)Γ(1+α).(ii) For 1<q<∞, we have
(4.8)∥G∥A˙q(ℝn)→A˙q(ℝn)=1.Since the dual space ofHA˙q(1<q<∞) is isomorphic to CM˙Oq' (see [18, 19]), Theorem 2.3 implies the following result.Theorem 4.3.
Let1<q<∞. Then Gω is a bounded operator on HA˙q(ℝn) if and only if (4.5) holds. Moreover, when (4.5) holds, the operator norm of Gω on HA˙q(ℝn) is given by
(4.9)∥Gω∥HA˙q(ℝn)→HA˙q(ℝn)=𝔼.Corollary 4.4.
For1<q<∞, we have
(4.10)∥G∥HA˙q→HA˙q=1.Following the idea in Section3, we define the higher order commutator of the weighted Cesàro operator as
(4.11)Gωb→f(x)=∫01‍(∏j=1k‍(bj(xt)-bj(x)))f(xt)t-nω(t)dt,x∈ℝn.
When k=0, Gωb→ is understood as Gω. Notice that if k=1, then Gωb→=Gωb. Similar to the proofs of Theorems 3.1 and 3.3, we have the following result.Theorem 4.5.
Letk≥2, 1<q1<q,q2,…,qk<∞, 1/q1=1/q+∑i=2k‍1/qi, -1/q<λ<0, -1/q1<λ1<0, 0≤λ2,…,λk<1/n, and λ1=λ+∑i=2k‍λi.
(i) Assume further thatω is a positive integrable function on [0,1]. The commutator Gωb→ is bounded from B˙q,λ(ℝn) to B˙q1,λ(ℝn), for any b→=(b2,…,bk)∈CM˙Oq2(ℝn)×⋯×CM˙Oqk(ℝn), if and only if
(4.12)∫01‍t-n(λ+1)ω(t)(log2t)k-1dt<∞.
(ii) Letλ2,…,λk>0 and b→=(b2,…,bk)∈CM˙Oq2,λ2(ℝn)×⋯×CM˙Oqk,λk(ℝn). Then the corresponding commutator Gωb→ is bounded from B˙q,λ(ℝn) to B˙q1,λ1(ℝn), provided that
(4.13)∫01‍t-n(λ+1)ω(t)dt<∞.We conclude this paper with some comments on the discrete version of the weighted Hardy and Cesàro operators.Letℕ0 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 ℝn. The discrete weighted Hardy operatorH~ω is defined by
(4.14)(H~ωf)(x)=∑k=0∞‍2-kf(2-kx)ω(2-k),x∈ℝn,
and the corresponding discrete weighted Cesàro operator is defined by setting, for all x∈ℝn,
(4.15)(G~ωf)(x)=∑k=0∞‍f(2kx)2k(n-1)ω(2-k).
We remark that, by the same argument as above with slight modifications, all the results related to the operators Hω and Gω in Sections 1–4 are also true for their discrete versions H~ω and G~ω.



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*Source: 780132-2012-09-12.xml*