# Weighted Central BMO Spaces and Their Applications

Authors: Huan Zhao; Zongguang Liu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, the central BMO spaces with MuckenhouptAp weight is introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on weighted Lebesgue spaces. The boundedness of vector-valued commutators on weighted Herz spaces is also considered.


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## Body

## 1. Introduction

For1<p<∞ and a nonnegative locally integrable function ω on ℝn, it is said that ω is in the Muckenhoupt Ap class if it satisfies the condition
(1)ωAp≔supQ1Q∫Qωxdx1Q∫Qωx−1/p−1dxp−1<∞.A weight functionω belongs to the class A1 if
(2)ωA1≔supQ1Q∫Qωxdxesssupx∈Qωx−1<∞.A weightω is called an A∞ weight if
(3)ωA∞≔supQ1Q∫Qωxdxexp1Q∫Qlogωx−1dx<∞.It is well-known thatA∞=⋃1≤p<∞Ap. Let ω∈A∞ and p∈0,∞; we denote Lpω as the space of all measurable functions f such that
(4)fLpω≔∫ℝnfxpωxdx1/p<∞.The definition ofAp weight was introduced by Muckenhoupt [1]. Weighted inequalities arise naturally in Fourier analysis, but their use is best justified by the variety of applications in which they appear. For example, the theory of weights plays an important role in the study of boundary value problems for the Laplace equation on Lipschitz domains. Other applications of weighted inequalities include vector-valued inequalities, extrapolation of operators, and applications to certain classes of integral equations and nonlinear partial differential equations. There are a number of classical results which demonstrate that the Muckenhoupt Ap classes are the right collections of weights to do harmonic analysis on weighted spaces. The main results along these lines are the equivalence between the ω∈Ap condition and the Lpω boundedness (or weak boundedness) of maximal operator and singular integral operators.A well-known result of Muckenhoupt [1] showed that the Hardy-Littlewood maximal operator M, that is
(5)Mfx=supQ∋x1Q∫Qfydy,is (weak) bounded on weighted Lebesgue spaces Lpω if and only if ω∈Ap for 1<p<∞ (for the case n=1). Hunt et al. [2] proved that the Ap condition also characterizes the Lpω boundedness of the Hilbert transform H, where
(6)Hfx=1πp.v.∫ℝfyx−ydy.Later, Coifman and Fefferman [3] extended the Ap theory to the case n≥1 and the general Calderón-Zygmund operators; they also proved that Ap weights satisfy the crucial reverse Hölder condition.On the other hand, it is well-known thatBMOℝn is just the dual space of Hardy space H1ℝn. Like this, the dual space of Herz-type Hardy space is the so-called central BMO space which is defined by
(7)CBMOpℝn=f∈Llocpℝn:fCBMOpℝn<∞,with
(8)fCBMOpℝn=supr>01B0,r∫B0,rfx−fB0,rpdx1/p,where
(9)fB0,r=1B0,r∫B0,rfxdx.The spaceCBMOpℝn can be regarded as a local version of BMOℝn at the origin, that is, BMOℝn⊊CBMOpℝn for 1≤p<∞ (see [4]). However, they have quite different properties. For example, there is no analysis of the famous John-Nirenberg inequality of BMOℝn for CBMOpℝn. See also [5–9] and [10] for more details. In 2007, Fu et al. [11] characterized CBMOpℝn space in terms of the boundedness of commutators of the Hardy operator.In this paper, we will introduce the space of centralBMO with Muckenhoupt Ap weight and characterize these spaces by the boundedness of commutator of the Hardy operator and its dual operator on weighted Lebesgue spaces. The boundedness of vector-valued commutators on weighted Herz spaces is also considered.Throughout this paper, the letterC denotes constants which are independent of the main variables and may change from one occurrence to another. Denote Bk=x∈ℝn:x≤2k and Ck=Bk\Bk−1, and χk is the characteristic function for k∈ℤ.


## 2. Weighted CentralBMO Spaces

In this section, we will introduce the definition of weighted centralBMO spaces and give some properties of CBMOpω.Let1≤p<∞, and ω is a nonnegative locally integrable function. A function f∈Llocpℝn is said to belong to the weighted central BMO spaces, if
(10)fCBMOpω≔supr>0f−fB0,rχB0,rLpωχB0,rLpω<∞.Whenω≡1 is a constant, CBMOpω is just CBMOpℝn.We recall some properties of the weighted Lebesgue spaces. LetYn denote the set of all families of disjoint and open cubes in ℝn. In [12], Diening et al. obtained the following lemma in the general case on Musielak-Orilicz spaces. But we only describe the special case on the weighted Lebesgue spaces now.Lemma 1.
Ifω∈A∞, then there exist 0<δ<1 and C>0 which only depend on the A∞-constant of ω such that
(11)∑Q∈QtQffQδχQLpω≤C∑Q∈QtQχQLpω,for all Q∈Yn; all tQQ∈Q,tQ≥0; and all f∈Lloc1ℝn with fQ≠0,Q∈Q.Lemma 2 (see [1]).
Letω∈Ap, 1≤p<∞; then, there exist constants C1,C2,δ>0 such that for all balls B in ℝn and all measurable subsets E⊂B,
(12)C1EBp≤ωEωB≤C2EBδ.
In fact, the first inequality of Lemma2 can be improved as follows.Lemma 3.
Letω∈Ap, 1<p<∞. Then, there exist p0 with 1<p0<p and C>0 such that for all balls B in ℝn and all measurable subsets E⊂B,
(13)χBLpωχELpω≤CBE1/p0.Proof.
By the fact thatAp=∪q<pAq and ω∈Ap (see [13]), there exist 1<p0<p such that ω∈Ap/p0. Applying Lemma 2, there exists a constant C>0 such that for any ball B and any measurable set E⊂B,
(14)EBp/p0≤CωEωB.
That is,(15)χBLpωχELpω≤CBE1/p0.
Therefore we have proved Lemma3.☐Now, we show the relationship betweenCBMOpω and central BMO spaces.Proposition 4.
Ifω∈Ap and 1<p<q<∞, then CBMOqω⊂CBMOpω⊊̈CBMOℝn.Proof.
Letf∈CBMOqω. For any B≔B0,r, by Hölder’s inequality, we have
(16)1ωB∫Bfx−fBpωxdx1/p≤1ωB∫Bfx−fBqωxdx1/q≤CfCBMOqω.
On the other hand, Letf∈CBMOpω, For any B≔B0,r, by Hölder’s inequality and the condition ω∈Ap, we have
(17)1B∫Bfx−fBdx≤1B∫Bfx−fBpωxdx1/p1B∫Bωx−p′/pdx1/p′≤CfCBMOpωωBB1/p1B∫Bωxdx−1/p≤CfCBMOpω.
Therefore, we only need to prove that there exists a functionf such that f∈CBMOℝn\CBMOpω. Without loss of generality, we may assume that n=1.
LetAk=x∈ℝ:2k<x≤2k+1,k∈ℤ+. Taking fx=∑k=0∞2kχAkxsgnx, then for any B≔B0,r,
(18)fB=1B∫Bfxdx=0.
Whenr≤1, we have fx≡0 and
(19)sup0<r≤11B∫Bfx−fBdx=0.
Whenr>1, there exists k0∈ℤ+ such that 2k0<r≤2k0+1; then,
(20)supr>11B∫Bfx−fBdx≤supr>1C2−k0∑k=0k0+1∫Ak2kdx≤C.
From (19) and (20), it follows that f∈CBMOℝn.
Whenr>4, there exists k0∈ℤ+ with k0≥2 such that 2k0<r≤2k0+1; then,
(21)fx−fBχBx≥∑k=0k0−12kχAkx≥2k0−1χAk0−1x≥CrχAk0−1x,which implies that
(22)supr>0f−fBχBLpωχBLpω≥supr>4f−fBχBLpωχBLpω≥supr>4CrχAk0−1LpωχBLpω.
SinceAk0−1⊂B, by Lemma 3, there exists p0 with 1<p0<p; we have
(23)supr>0f−fBχBLpωχBLpω≥supr>4CrBAk0−1−1/p0=supr>4Cr1−1/p0=∞.
Therefore,f∉CBMOpω.☐Proposition 5.
Ifω∈Ap and 1<p<∞, then there exists a constant q>p such that CBMOqℝn⊂CBMOpω.Proof.
We can take a cubeQB so that B⊂QB⊂nB. By Lemma 1, there exists a constant 0<δ<1 independent of B such that for all f∈Lloc1ℝn,
(24)fδχQBLpω≤CfQBδχQBLpω.
Letq=1/δ and f=b−bBqχB. We conclude that
(25)fQBδ=1QB∫Bbx−bBqdx1/q≤CbCBMOqℝn.
By Lemma3, there exists a constant 1<p0<p such that
(26)χQBLpω≤χnBLpωχBLpωχBLpω≤CnBB1/p0χBLpω≤CχBLpω.
This gives us(27)bCBMOpω≤CbCBMOqℝn.
Hence, the proof of Proposition5 is completed.☐Proposition 6.
Ifω∈Ap and 1<p<∞, then f∈CBMOpω if and only if there exist a collection of numbers cB0,rr>0 (i.e., for each ball B0,r, there exists cB0,r∈ℝ such that
(28)supr>0χB0,rLpω−1f−cB0,rχB0,rLpω<∞.Proof.
We setcB0,r=fB0,r for all balls B0,r; the necessity of the condition in Proposition 6 holds. Let us check the sufficiency of Proposition 6.
A similar argument as Proposition4, we have, for any B≔B0,r,
(29)1B∫Bfx−cBdx≤CχBLpω−1f−cBχBLpω.
Thus,(30)f−fBχBLpωχBLpω≤f−cBχBLpωχBLpω+cB−fBχBLpωχBLpω≤C+cB−fB≤C.
Therefore,f∈CBMOpω; the proof of Proposition 6 is completed.☐Proposition 7.
Ifω∈Ap and 1<p<∞, then f∈CBMOpω if and only if
(31)fCBMO∗pω≔supr>0infc∈ℂχB0,rLpω−1f−cχB0,rLpω<∞.Proof.
The proof of Proposition7 is similar as that of Proposition 6; we omit the details.☐


## 3. Characterization ofCBMOpω Spaces via Commutators

We first review the definitions of then-dimensional Hardy operator and its dual operator. For a locally integrable function f in ℝn, the n-dimensional Hardy operator H is defined by
(32)Hfx=1xn∫∣y∣≤xfydy,x∈ℝn\0.The dual Hardy operatorH∗ is defined by
(33)H∗fx=∫∣y∣>xfyyndy,x∈ℝn\0.Letb be a locally integrable function on ℝn. The commutators of H and H∗ are defined by
(34)b,Hf=bHf−Hbfand(35)b,H∗f=bH∗f−H∗bf.The study of the Hardy operator has a very long history, and a number of papers involved its generalizations, variants, and applications. For the earlier development of this kind of integrals and many important applications, we refer the interested reader to the masterpiece [14]. We are interested in the characterization of commutator of the Hardy operator.Now, we give a remarkable result about the commutator of the Hardy operator; that is, Fu et al. [11] showed the following.Theorem 8.
Let1<p<∞, 1/p+1/p′=1, and b∈CBMOmaxp,p′ℝn. Then, both b,H and b,H∗ are bounded operators on Lpℝn. Conversely,
(i)
ifb,H is a bounded operator on Lpℝn, then b∈CBMOp′ℝn(ii)
ifb,H∗ is a bounded operator on Lpℝn, then b∈CBMOpℝnThe following consequence improves Theorem8.Theorem 9.
Ifω∈Ap, 1<p<∞ and μ=ω1−p′. Then, the following statements are equivalent:
(i)
b∈CBMOpω∩CBMOp′μ(ii)
b,H and b,H∗ are bounded from Lpω to LpωProof.
i⇒ii. We focus on the proof of the boundedness of b,H, since the arguments of b,H∗ are similar with necessary modifications.
Forf∈Lpω, we have
(36)b,HfLpω=∑k=−∞∞χkb,HfLpω=∑k=−∞∞χk··n∫B0,∣·∣b·−byfydyLpω≤∑k=−∞∞χk·∑j=−∞k1·n∫Cjb·−byfydyLpω.
It is easy to see that(37)∫Cjbx−byfydy≤∫Cjbx−bBkfydy+∫CjbBk−byfydy.
By Hölder’s inequality, we get(38)∫Cjbx−bBkfydy≤Cbx−bBk∫Cjfypωydy1/p∫Cjωy−p′/pdy1/p′≤Cbx−bBkfχjLpωχjLp′μ.
In [11], Fu et al. showed that for b∈CBMOℝn and j,k∈ℤ,
(39)bt−bBk≤bt−bBj+Cj−kbCBMOℝn.
By Proposition4, for b∈CBMOpω⊂CBMOℝn and j,k∈ℤ, we have
(40)bt−bBk≤bt−bBj+Cj−kbCBMOpω.
This gives us(41)∫Cjby−bBkfydy≤∫Cjby−bBjfydy+bBk−bBj∫Cjfydy≤∫Cjby−bBjfyωy1pωy−1pdy+Ck−jbCBMOpω∫Cjfyωy1pωy−1pdy≤Cb−bBjχjLp′μfχjLpω+Ck−jbCBMOpωfχjLpωχjLp′μ≤CbCBMOp′μfχjLpωχBjLp′μ+Ck−jbCBMOpωfχjLpωχjLp′μ≤Ck−jfχjLpωχBjLp′μ.
Combing (38) and (41), we get
(42)χk·1·n∫Cjb·−by∣fy∣dyLpω≤C2−knb−bBkχkLpωfχjLpωχjLp′μ+C2−knk−jχkLpωfχjLpωχBjLp′μ≤C2−knk−jχBkLpωfχjLpωχBjLp′μ.
From the conditionω∈Ap and Lemma 2, it follows that for k≥j, there exists a constant δ∈0,1 such that
(43)2−knχBkLpωχBjLp′μ≤CχBjLp′μχBkLp′μ≤CBjBkδ/p′≤C2j−knδ/p′.
Therefore, generalized Minkowski’s inequality implies(44)b,HfLpω≤C∑k=−∞∞∑j=−∞kk−j2j−knδ/p′fχjLpω≤C∑j=−∞∞∑k=j∞k−j2j−knδ/p′fχjLpω≤CfLpω.
ii⇒i. The condition b∈CBMOpω∩CBMOp′μ turns out to be necessary for the conclusion that both b,H and b,H∗ are bounded on Lpω.
For any ballB≔B0,r and x∈B, we have
(45)bx−bB=1B∫Bbx−bydy≤C1xn∫y≤xbx−byχBydy+C∫x≤ybx−byχByynB−1yndy≤Cb,HχBx+Cb,H∗f0x,where f0=xnB−1χBx.
Fromb,H and b,H∗ that are bounded on Lpω, it follows that
(46)b−bBχBLpω≤Cb,HχBLpω+Cb,H∗f0Lpω≤CχBLpω+Cf0Lpω≤CχBLpω.
Therefore, we obtain thatb belongs to CBMOpω.
Note thatLpω′=Lp′ω1−p′ (see [15]). We know that b,H and b,H∗ are bounded on Lp′μ. Therefore, we obtain that b∈CBMOp′μ.
This completes the proof of Theorem9.☐


## 4. Vector-Valued Inequality

In this section, we give the definition of weighted Herz spaces ([16]). Let α∈ℝ, 0<p,q<∞, and ω be weight functions on ℝn. The homogeneous weighted Herz space K̇pα,qω is defined by
(47)K̇pα,qω≔f∈Llocpℝn\0:fK̇pα,qω<∞,where
(48)fK̇pα,qω≔2αkfχkLpωk=−∞∞ℓq.We prove the boundedness of the vector-valued commutator of the Hardy operator on weighted Herz spaces.Theorem 10.
Letω∈Ap, 1<r,p<∞, 0<q<∞, and b∈CBMOpω∩CBMOp′μ.
(i)
Ifα<n/p′, then there exists a constant C such that(49)∑j=1∞b,Hfjr1/rK̇pα,qω≤C∑j=1∞fjr1/rK̇pα,qω,for all sequences of functions fjj=1∞ satisfying ∥∥fjj∥ℓr∥K̇pα,qω<∞(ii)
Ifα>−n/p, then there exists a constant C such that(50)∑j=1∞b,H∗fjr1/rK̇pα,qω≤C∑j=1∞fjr1rK̇pα,qω,for all sequences of functions fjj=1∞ satisfying fjjℓrK̇pα,qω<∞In order to prove Theorem10, we additionally introduce the next lemma well-known as the generalized Minkowski inequality.Lemma 11.
If1<r<∞, then there exists a constant C>0 such that for all sequences of functions fjj=1∞ satisfying fjjℓrK̇pα,qω<∞,
(51)∑j=1∞∫ℝnfjydyr1/r≤C∫ℝn∑j=1∞fjyr1/rdy.Proof of Theorem10.
We focus on the proof of the boundedness ofb,H, since the arguments of b,H∗ are similar with necessary modifications. For every fjj=1∞ with fjjℓrK̇pα,qω<∞, we obtain
(52)b,HfjjℓrK̇pα,qω=∑k=−∞∞2αqkχkb,H∑l=−∞∞fjχljℓrLpωq1/q≤∑k=−∞∞2αqkχk∑l=−∞kb,HfjχljℓrLpωq1/q.
For convenience, below, we denoteF≔∥fjj∥ℓr. For x∈Ck, generalized Hölder’s inequality and generalized Minkowski’s inequality (51) imply
(53)b,Hfjχlxjℓr≤C1xn∫Clbx−byfjydyjℓr≤C2−kn∫Clbx−byfjydyjℓr≤C2−kn∫Clbx−byFydy≤C2−knbx−bBl∫ClFydy+∫ClbBl−byFydy≤C2−knFχlLpωbx−bBlχlLp′μ+b−bBlχlLp′μ,from the fact that
(54)bx−bBl≤bt−bBk+Cl−kbCBMOpω,which gives us
(55)χkb,HfjχljℓrLpω≤C2−knFχlLpω×k−lbCBMOpωχkLpωχlLp′μ+bCBMOp′μχBlLp′μχkLpω≤Ck−lFχlLpω2−knχBlLp′μχkLpω≤Ck−lFχlLpω2−knχkLpωBlχBlLpω−1.
By Lemma3 and α/n<1/p′, there exists a constant 1<p0<p such that α/n<1/p′0 and
(56)χkLpωχBlLpω≤χBkLpωχBlLpω≤C2nk−l/p0.
Then,(57)χkb,HfjχljℓrLpω≤Ck−l2nl−k/p′0FχlLpω.
This implies that(58)b,HfjjℓrK̇pα,qω≤C∑k=−∞∞2αqk∑l=−∞kk−l2nl−k/p0′FχlLpωq1/q=C∑k=−∞∞∑l=−∞k2αlk−l2l−kn/p0′−αFχlLpωq1/q.
If0<q≤1, then we obtain
(59)b,HfjjℓrK̇pα,qω≤C∑k=−∞∞∑l=−∞k2αqlFχlLpωq2l−kn/p0′−αqk−lq1/q≤C∑l=−∞∞2αqlFχlLpωq∑k=l∞2l−kn/p0′−αqk−lq1/q≤C∑l=−∞∞2αqlFχlLpωq1/q≤CFK̇pα,qω.
If1<q<∞, then we use Hölder’s inequality and obtain
(60)b,HfjjℓrK̇pα,qω≤C∑k=−∞∞∑l=−∞k2αqlFχlLpωq2l−kn/p0′−αq/2×∑l=−∞kk−lq′2l−kn/p0′−αq′/2q/q′1/q≤C∑k=−∞∞∑l=−∞k2αqlFχlLpωq2l−kn/p′0−αq/21/q≤C∑l=−∞∞2αqlFχlLpωq∑k=l∞2l−kn/p′0−αq/21/q≤C∑l=−∞∞2αqlFχlLpωq1/q≤CFK̇pα,qω.
This completes the proof of Theorem10.



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