# BMO Functions Generated byAXℝn Weights on Ball Banach Function Spaces

Authors: Ruimin Wu; Songbai Wang
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

LetX be a ball Banach function space on ℝn. We introduce the class of weights AXℝn. Assuming that the Hardy-Littlewood maximal function M is bounded on X and X′, we obtain that BMOℝn=αlnω:α≥0,ω∈AXℝn. As a consequence, we have BMOℝn=αlnω:α≥0,ω∈ALp·ℝnℝn, where Lp·ℝn is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space Xω for any ω∈AXℝn, then the commutator b,T is bounded on X with b∈BMOℝn.


---

## Body

## 1. Introduction

It is well known that there is a relation betweenA∞ℝn weights and BMOℝn, i.e., for any p∈1,∞,
(1)BMOℝn=αlnW:α≥0,W∈Apℝn.See, for instance, [1] (p. 409). The purpose of this note is to reveal the relation between BMOℝn and AXℝn weights over the ball Banach function space X.To state our results, we begin with the definition of the ball Banach function space. Denote by the symbolMℝn the set of all measurable functions on ℝn. For any x∈ℝn and r∈0,∞, let Bx,r≔y∈ℝn:x−y<r and
(2)B≔Bx,r:x∈ℝnandr∈0,∞.Definition 1.
A Banach spaceX⊂Mℝn is called a ball Banach function space if it satisfies that
(i)
fX=0 implies that f=0 almost everywhere(ii)
g≤f almost everywhere implies that gX≤fX(iii)
0≤fm↑f almost everywhere implies that fmX↑fX(iv)
B∈B implies that 1B∈X, where B is as in (2);(v)
for anyB∈B, there exists a positive constant CB, depending on B, such that, for any f∈X,(3)∫B∣fx∣dx≤CBfXFor any ball Banach function spaceX, the associate space (Köthe dual) X′ is defined by setting
(4)X′≔f∈Mℝn:fX′≔sup∥fg∥L1ℝn:g∈X,∥g∥X=1<∞,where ·X′ is called the associate norm of ·X (see, for instance, [2] (Chapter 1, Definitions 2.1 and 2.3)).Remark 2.
By [3] (Proposition 2.3), we know that, ifXis a ball Banach function space, then its associate spaceX′is also a ball Banach function space.Now, we introduce the class of weightsAXℝn and recall the function space BMO. A weight ω is a locally integrable function such that 0<ωx<∞ almost everywhere x∈ℝn.Definition 3.
LetXbe a ball Banach function space. We say that a weightωbelongs toAXℝnif(5)supB∈B∥ω1B∥X∥1B∥X∥ω−11B∥X′∥1B∥X′<∞,here and hereafter 1B is the characteristic function for B.Remark 4.
(1)
There is an immediate consequence. LetX be a ball Banach function space. If ω∈AXℝn, then ω−1∈AX′ℝn(2)
We recall that the definition ofApℝn. Let p∈1,∞. A weight W belongs to Apℝn if
(6)supB∈B1B∫BWxdx1B∫BWx1−p′dxp−1<∞By the definition ofAXℝn and Apℝn, W∈Apℝn if and only if ω≔W1/p∈ALpℝnℝn for any p∈1,∞.The classical function spaceBMOℝn is the collection of all locally integrable functions f such that
(7)BMOℝn≔supB1B∫Bfx−fBdx,where the supremum is taking all balls B in ℝn and fB is the mean value of the function f on B, namely,
(8)fB≔1B∫Bfydy.By the well-known John-Nirenberg inequality, John and Nirenberg [4] proved that there exists a positive constant C such that
(9)∥f∥BMOℝn≤∥f∥BMOLpℝn≤C∥f∥BMOℝn,where p∈1,∞ and
(10)BMOLpℝn≔supB1B∫Bfx−fBpdx1/p.We also recall that the Hardy-Littlewood maximal functionM is defined by setting, for any locally integrable function f and x∈ℝn,
(11)Mfx≔supB∈B1BxB∫Bfydy.Now, we state our result as the following theorem.Theorem 5.
LetXbe ball Banach function spaces. If the Hardy-Littlewood maximal functionMis bounded onXandX′, then(12)BMOℝn≔αlnω:α≥0,ω∈AXℝn.Remark 6.
Letp∈1,∞, Theorem5goes back to the classical result forX≔Lpℝn.As an example, letP=Pℝn be the collection of all measurable functions p·:ℝn⟶1,∞. Then, the variable Lebesgue space Lp·ℝn is defined to be the set of all measurable functions f on ℝn such that
(13)fLp·ℝn≔infλ∈0,∞:∫ℝnfxλpxdx≤1<∞.Denotep−≔essinfx∈ℝnpx and p+≔esssupx∈ℝnpx. A measurable function p·∈Pℝn is said to be globally log-Hölder continuous if there exists a p∞∈ℝ such that, for any x,y∈ℝn,
(14)px−py≲1loge+1/x−y,px−p∞≲1loge+∣x∣,where the implicit positive constants are independent of x and y.Definition 7 ([5], Definition 1.4.).
Given an exponent functionp·:ℝn⟶1,∞and a weightω, we say thatω∈Ap·if there exists a constantKsuch that for every ballB,
(15)ω1BLp·ℝnω−11BLp·′ℝn≤CB,where 1/px+1/px′=1 for almost everywhere x∈ℝn.Remark 8.
Letp·be a globally log-Hölder continuous function satisfying1<p−≤p+<∞. By [3] (Lemma 2.5 and Proposition 3.8.), for any ballB⊂ℝn, ∣B∣=∥1B∥Lp·ℝn∥1B∥Lp·′ℝn. This shows that forX=Lp·ℝn(16)ALp·ℝnℝn=Ap·ℝn.Letp· be a globally log-Hölder continuous function satisfying 1<p−≤p+<∞. We know that M is bounded on Lp·ℝn and its duality Lp·′ℝn; see, for instance, [6, 7] and their references.Corollary 9.
Letp·be a globally log-Hölder continuous function satisfying1<p−≤p+<∞. Then,BMOℝn=αlnω:α≥0,ω∈Ap·ℝn.


## 2. Proof of Theorem5

The following lemmas give two elementary properties of ball Banach function spaces, whose proof is similar to the one corresponding to Banach function spaces; see [2].Lemma 10 (Holder’s inequality).
LetXbe a ball Banach function space with the associate spaceX′. Iff∈Xandg∈X′, thenfgis integrable and(17)∫ℝn∣fxgx∣dx≤∥f∥X∥g∥X′.Lemma 11 (G. G. Lorentz, W. A. J. Luxembourg).
Every ball Banach function spaceXcoincides with its second associate spaceX″. In other words, a functionfbelongs toXif and only if it belongs toX″and, in that case,(18)fX=fX″.Under weak boundedness of the Hardy-Littlewood maximal functionM on X, the norm ∥·∥X enjoys the following property; see [8] (Lemma 2.2).Lemma 12.
LetXbe a ball Banach function space and suppose that the Hardy-Littlewood maximal operatorMis weakly bounded onXorX′, that is, there exists a positive constantCsuch that(19)1x∈ℝn:Mfx>λX≤Cλ−1fXor
(20)1x∈ℝn:Mfx>λX,≤Cλ−1fX′holds for all λ>0 and all f∈X. Then, there exists a positive constant C such that for all balls B∈B, ∥1B∥X∥1B∥X′≤CB.Remark 13.
By Lemma10, we haveB≤1BX1BX′for any ballB∈B.Lemma 14.
Letφ∈Lloc1ℝnandXbe a ball Banach function space. Suppose that the Hardy-Littlewood maximal operatorMis weakly bounded onX. Then,eφ∈AXℝnif and only if there exists a positive constantCsuch that for any ballB∈B(21)eφ−φB1BX1BX≤C,e−φ−φB1BX′1BX′≤C.Proof.
We first prove the sufficiency. In fact, by the definition ofAXℝn, we have
(22)eφ1BX1BXe−φ1BX′1BX′=eφ−φB1BX1BXe−φ−φQ1BX′1BX′≤C.
Conversely, suppose thateφ∈AXℝn. Then by Lemmas 10 and 12(23)eφ−φB1BX1BX=e−φQeφ1BX1BX≤1B∫Be−φxdxeφ1BX1BX≤e−φX′1BXBeφ1BX1BX≤C.
Also,(24)e−φ−φB1BX′1BX′=eφQe−φ1BX′1BX′≤1B∫Beφxdxeφ1BX′1BX′≤eφX1BX′Be−φ1BX′1BX′≤C.The John-Nirenberg inequality for ball Banach function spacesX was established by Izuki et al. ([9], Theorem 3.1).Lemma 15.
LetXbe a ball Banach function space such thatMis bounded onX′and writeC0≔∥M∥X′→X′. Then, there exists a positive constantC1such that for all ballsB, f∈BMOℝnandλ≥0,
(25)1x∈B:∣f−fB∣>λX≤C12−λ/2n+21+2n+4C0∥f∥BMO2n+21+2n+4C0∥f∥BMO1BX.As a consequence of Lemma15, we have the following inequality.Lemma 16.
LetXbe a ball Banach function space. Suppose thatMis bounded onX′. Suppose thatφ∈BMO. Then for anyα∈0,ln2/2n+21+2n+4C0∥f∥BMOand ballB∈B, we have(26)eα∣φ−φB∣1BX≤C11−2α/ln2−1/2n+21+2n+4C0∥f∥BMO21/2n+21+2n+4C0∥f∥BMO1BX,where C1 is as in Lemma 15.Proof.
By Lemma15, we have
(27)eα∣φ−φB∣X≤∑k=0∞1x∈B:k+1≥∣φ−φB∣>keα∣φ−φB∣X≤∑k=0∞eαk+11x∈B:∣φ−φB∣>kX≤∑k=0∞C1eαk+12−k/2n+21+2n+4C0∥f∥BMO∥1B∥X=∑k=0∞C1eα2αk/ln2−k/2n+21+2n+4C0∥f∥BMO∥1B∥X≤C11−2α/ln2−1/2n+21+2n+4C0∥f∥BMO21/2n+21+2n+4C0∥f∥BMO∥1B∥X.Lemma 17.
LetXbe a ball Banach function space. Ifω∈AXℝn, thenlnω∈BMOℝn.Proof.
Letφ≔lnω. Then, ω=eφ. By Lemmas 10, 12, and 14, we obtain that
(28)∥φ∥BMOℝn≤CsupB∈B1∣B∣∫Be∣φx−φB∣dx≤CsupB∈B1∣B∣∫x∈B:φx−φB≥0eφx−φBdx+∫x∈B:φx−φB<0e−φx−φBdx≤CsupB∈B1∣B∣∥eφ−φB1B∥X∥1B∥X′+∥e−φ−φB1B∥X′∥1B∥X<∞.Proof of Theorem 18.
By Lemma17, for any ω∈AXℝn and α≥0, φ≔αlnω∈BMOℝn. Conversely, suppose that φ∈BMOℝn. Since M is bounded on X′, by Lemma 16, we know that there exist β1∈0,ln2/2n+21+2n+4∥M∥X′→X′∥φ∥BMO and C3∈0,∞ such that, for any ball B∈B,
(29)∥eβ1∣φ−φB∣1B∥X∥1B∥X≤C3.Similarly, sinceM is bounded on X, by Lemmas 11 and 16, we know that there exist β2∈0,ln2/2n+21+2n+4∥M∥X→X∥φ∥BMO and C4∈0,∞ such that, for any ball B∈B,
(30)∥eβ2∣φ−φB∣1B∥X′∥1B∥X′≤C4.Takingα=minβ1,β2 and C=maxC3,C4 and applying Lemma 14, we get the desired result.


## 3. Applications

In this section, we will show that the boundedness of the commutator of a linear operatorT on X with the BMO function can be derived from the weighted boundedness of T on X. We first establish the following Minkowski-type inequality.Lemma 19.
LetXbe a Banach function space andFa measurable function onℝn×ℝm. If, for almost everyx∈ℝn, Fx,·∈L1ℝmand, for almost everyy∈ℝm, F·,y∈X, then(31)∫ℝm∣F·,y∣dyX≤∫ℝmF·,yXdy.Proof.
By Lemma11, we have
(32)∫ℝm∣F·,y∣dyX=∫ℝm∣F·,y∣dyX″=sup∫ℝn∫ℝm∣Fx,y∣dygxdx:g∈X′suchthat∥g∥X′=1.
From the Fubini theorem and Lemma10, it follows that
(33)∫ℝn∫ℝmFx,ydygxdx≤∫ℝn∫ℝmFx,ygxdydx=∫ℝm∫ℝnFx,ygxdxdy≲∫ℝmF·,yXgX′dy=∫ℝmF·,yXdy,which implies the desired conclusion. This finishes the proof of Lemma 19.LetT be a linear operator defined by
(34)Tfx≔∫ℝnKx,yfydy.Given a symbolb, we define the commutator b,Tf=bTf−Tbf.Letω be a weight. Define Xω≔f∈Mℝn:fω∈X and ∥f∥Xω≔∥fω∥X. We say that T is bounded on Xω if there exists a positive constant C such that for all f∈Xω,
(35)TfωX≤C∥fω∥X.Theorem 20.
LetXbe a ball Banach function space. Suppose thatMis bounded onXandX′. If, for anyω∈AXℝn, Tis bounded onXωthen, for allb∈BMOℝn, b,Tis bounded onX, i.e.,(36)b,TfX≤C∥f∥X,where C is independent of f.Proof.
We adapt the idea from [10, 11]. Without loss of generality, we assume that b≠0 in BMOℝn. By Theorem 5, there exists a α∈0,∞ such that eαb∈AXℝn. As well known, for every θ∈0,2π, bcosθ∈BMOℝn and ∥bcosθ∥BMOℝn=∥b∥BMOℝn. Thus,
(37)eαbcosθ∈AXℝn.
For anyz∈ℂ, define gz≔ezαbx−by. Then, gz is analytic on ℂ and the Cauchy integral formula implies that
(38)bx−by=g′0α=12απi∫∣z∣=1gzz2dz=12απ∫02πeiθαbx−bye−iθdθ.
For anyθ∈0,2π, set hθx≔fxe−αbxeiθ. Since f∈X, we have
(39)hθXeαbcosθ=fxe−αbeiθeαbcosθX=fX.
By this and (38), we conclude that
(40)b,Tfx=∫ℝnKx,y12π∫02πeiθαbx−bye−iθdθfydy=12π∫02πThθxeαbxeiθe−iθdθ.
Applying Lemma19 and the weighted boundedness of T, we have
(41)b,TfX≤12απ∫02πThθXeαbcosθdθ≤CfX.
We complete the proof of Theorem20.



---

*Source: 6626787-2021-01-08.xml*