Let T be the unite circle on $R^2$. Denote by BMO(T) the classical BMO space
and denote by BMO_D(T) the usual dyadic BMO space on T. We prove that, BMO(T)
is the intersction of BMO_D(T) and a translate of BMO_D(T).Comment: 4 page

Mei, Tao

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eC. R. Acad. Sci. Paris, Ser. I 336 (2003) 1003–1006Harmonic Analysis/Functional AnalysisBMO is the intersection of two translates of dyadic BMOTao MeiMathematics Department, Texas A&M University, College Station, TX 77843, USAReceived 24 February 2003; accepted after revision 29 April 2003Presented by Gilles PisierAbstractLet T be the unit circle onR2. Denote by BMO(T) the classical BMO space and denote by BMOD(T) the usual dyadicBMO space onT. Then, for suitably chosenδ ∈ R, we have‖ϕ‖BMO(T)  ‖ϕ‖BMOD(T) + ‖ϕ(· − 2δπ)‖BMOD(T), ∀ϕ ∈ BMO(T).To cite this article: T. Mei, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.RésuméBMO est l’intersection de deux translatés de BMO dyadique. Soit T le cercle unité dansR2. On note BMO(T) l’espaceBMO classique et l’on note BMOD(T) l’espace BMO dyadique usuel surT. Pour certaines valeurs deδ ∈ R, nous montronsque l’espace BMO(T) coïncide avec l’intersection de BMOD(T) et du translaté parδ de BMOD(T), en d’autres termes qul’on a‖ϕ‖BMO(T)  ‖ϕ‖BMOD(T) + ‖ϕ(· − 2δπ)‖BMOD(T), ∀ϕ ∈ BMO(T).Pour citer cet article : T. Mei, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.1. IntroductionLet T be the unit circle onR2, identified with(0,2π]. Recall thatBMO(T) ={ϕ ∈ L1(T): ‖ϕ‖BMO(T) = sup{1|I |∫I|ϕ − ϕI |dθ}< ∞},where the supremum runs over all intervalsI on T andϕI = 1|I |∫Iϕ(s)ds. Let D = {Dn}n0 be the family of theusual dyadicσ -algebra onT, i.e.,Dn = σ{(Dkn)0k<2n}, Dkn =(2πk2−n,2π(k + 1)2−n]; n  0.E-mail address:tmei@math.tamu.edu (T. Mei).1631-073X/03/$ – see front matter 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rightsreserved.doi:10.1016/S1631-073X(03)00234-61004 T. Mei / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 1003–1006Thebetweenegntof theirximalRecall that the usual dyadic BMO space is defined byBMOD(T) ={ϕ ∈ L1(T): ‖ϕ‖BMOD(T) = supn,k{2n2π∫Dkn|ϕ − ϕDkn |dθ}< ∞}.BMO(T) and the dyadic BMO space BMOD(T) have many similarities, but nevertheless certain differences.dyadic BMO space is usually much easier to study. Some works have been done to study the relationshipthe two kinds of BMO spaces (see [1,4]). In this paper, we show that, for any positiveδ suitably chosen (morprecisely satisfyingd(δ) > 0, with d(δ) as defined below)ϕ is in BMO(T) if and only if ϕ(·) andϕ(· − 2πδ) arein BMOD(T). Clearly the analogous result holds onR with the same proof (see the final remark below).2. The main resultLet A be the collection of all dyadic rationals. For 0< δ < 1, define its relative distance toA, denoted byd(δ)in this paper, as followsd(δ) := inf{2n∣∣δ − k2−n∣∣ | n  0, k ∈ Z}.Let T be the unit circle onR2. For δ with d(δ) > 0, we consider the filtrationDδ = {Dδn}n0 on T obtained fromthe usual dyadic filtration after translation by 2πδ. More precisely:Dδn = σ{(Dδ,kn)0k<2n}, Dδ,kn =(2δπ + 2πk2−n,2δπ + 2π(k + 1)2−n], ∀n  0.Hence, if we define‖ϕ‖BMODδ (T) in the usual way, we have‖ϕ‖BMODδ (T) = ‖ϕ(· − 2δπ)‖BMOD(T).In this paper, we will sayD (resp.Dδ) “fits” an intervalI ⊂ T with fit-constantc if there existn  0, 0  kI <2n such thatI ⊂ DkIn (resp.I ⊂ Dδ,kIn ) and|DkIn |  c|I | (resp.|Dδ,kIn |  c|I |). Our key observation is the followinsimple fact.Proposition 2.1. For any intervalI ⊂ T, eitherD or Dδ fits I with fit-constant2/d(δ).Proof. If |I |  2πd(δ), let n = kI = 0, thenI ⊂ D00 = (0,2π].If |I | < 2πd(δ), let n  0 be the integer such thatd(δ)2π2−n−1  |I | < d(δ)2π2−n. SetAn ={k2π2−n; 0  k < 2n}, Aδn = {2δπ + k2π2−n; 0 k < 2n}.Note that for any two pointsa, b ∈ An ∪Aδn, we have|a −b|  d(δ)2π2−n > |I |. Thus there is at most one elemeof An ∪Aδn belonging toI. ThenI ∩An = φ or I ∩Aδn = φ. Therefore,I must be contained in someDkIn or Dδ,kInand|DkIn | = |Dδ,kIn | = 2π/2n  2/d(δ)|I |. ✷Remark 1. From the above proposition, a number of “classical” results become immediate consequences“probabilistic” counterparts. For instance, Doob’s maximal inequality implies the Hardy–Littlewood mainequality immediately.Theorem 2.2. For ϕ ∈ L1(T), 0 < δ < 1, with d(δ) > 0, we have‖ϕ‖BMO(T)  4d(δ)max{‖ϕ‖BMOD(T),∥∥ϕ(· − 2δπ)∥∥BMOD(T)}.Proof. By the above proposition, for every intervalI ⊂ T, there existN,kI such thatI ⊆ DkIN or I ⊆ Dδ,kIN and2πN  2d(δ) |I |. If Dδ,kIN containsI, then2T. Mei / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 1003–1006 1005tntase: seeoteesformsfourthen1|I |∫I∣∣ϕ(θ) − ϕI ∣∣dθ  1|I |∫I∣∣ϕ(θ) − ϕDδ,kIN∣∣dθ + |ϕDδ,kIN− ϕI |  2|I |∫I∣∣ϕ(θ) − ϕDδ,kIN∣∣dθ 4d(δ)|Dδ,kIN |∫Dδ,kIN∣∣ϕ(θ) − ϕDδ,kIN∣∣dθ  4d(δ)‖ϕ‖BMODδ(T) = 4d(δ)∥∥ϕ(· − 2δπ)∥∥BMOD(T).If DkIN containsI, then similarly1|I |∫I∣∣ϕ(θ) − ϕI ∣∣dθ  4d(δ)‖ϕ‖BMOD(T).Thus, taking the supremum over all intervalsI ⊂ T, we get‖ϕ‖BMO(T)  4d(δ)max{‖ϕ‖BMOD(T),∥∥ϕ(· − 2δπ)∥∥BMOD(T)}. ✷Example 1. Let δ = 1/3, thend(δ) = 1/3, and then‖ϕ‖BMO(T)  12 max{‖ϕ‖BMOD(T),∥∥∥∥ϕ(· − 2π3)∥∥∥∥BMOD(T)}.Remark 2. Let ϕ#(t) = supIt 1|I |∫I |ϕ − ϕI |dθ andϕ#D(t) = supDknt 1|Dkn|∫Dkn|ϕ − ϕDkn |dθ . It is easy to see tha{δ, d(δ) > 0} is exactly the set of allδ’s such thatϕ#  c max{ϕ#D, ϕ#D(·−2πδ)} for somec > 0. The same statemetrivially remains valid in the Banach space valued case and is particularly useful in the operator valued c[3] for some results in that direction.Remark 3. One can check that the set{δ, d(δ) > 0} is dense in(0,1) while its measure is zero.Corollary 2.3. BMO(T) = BMOD(T) ∩ BMODδ (T) with equivalent norms.Denote byH 1D (resp.H1Dδ ) the dyadic Hardy space with respect toD (resp.Dδ). By duality, we haveCorollary 2.4. H 1 = H 1D + H 1Dδ with equivalent norms.Remark 4. There is another way to see Corollary 2.4. Denote byH 1,at the classical atomic Hardy space. Denby H 1,atD (resp.H1,atDδ ) the dyadic atomic Hardy space with respect toD (resp.Dδ). From Proposition 2.1, we sethat any atom is a dyadic atom (up to a fixed factor) with respect to eitherD or Dδ . ThusH 1,at = H 1,atD + H 1,atDδwith equivalent norms. SinceH 1,at = H 1 andH 1,atD = H 1D, we obtain Corollary 2.4.Remark 5. See [4] for a recent result (of the same flavor) comparing Hilbert transforms and martingale tranproved by averaging shifted and dilated dyadic filtrations.Remark 6. John Garnett kindly informed us that he already knew that BMO(T) coincides with the intersection othree (suitably chosen) translates of dyadic BMO(T) (the idea for this can be traced back to p. 417 of [2]), butmain result seems new.We now turn to the case of dimensionm > 1. By a straightforward product argument, one can deduce fromabove proposition that BMO(Tm) coincides with the intersection of a family of 2m translates of the dyadic versioof BMO(Tm). However, we wish to show below that the number of translates can be reduced tom + 1.1006 T. Mei / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 1003–1006tf0 (2000)In the following, we always suppose{δi}mi=0 is a sequence in(0,1) such thatd({δi}mi=0) := mini =j d(δi − δj ) > 0.Let Dδi be the translation by 2πδi of the family of the usual (one dimensional) dyadicσ -algebra. SetF i =(Dδi )m,0  i  m. Then we getm + 1 families of increasing dyadicσ -algebras onTm.Proposition 2.5. LetF i , 0  i  m, be as above and letc = 2/d({δi}mi=0). Then, for any cubeJ ⊂ Tm, there existssomeF i which fitsJ with fit-constantcm.Proof. Write J ⊂ Tm asJ = J1 × J2 × · · · × Jm, whereJi are intervals inT, 1 i  m. Let {δi}mi=0 be such thad({δi}mi=0) > 0. By Proposition 2.1, for everyJi , there is at most oneki,0  ki  m, such thatDδki does not fitJiwith constantc. Then there is at least oneDδk which fits allJi with constantc. Thus (with an obvious extension oour terminology) we may say thatFk fits J with fit-constantcm. ✷From Proposition 2.5 we haveTheorem 2.6 (In the case ofTm). Let {δi}mi=0 be a sequence in(0,1) such thatd({δi}mi=0) > 0. Let iδ =(δi, δi , . . . , δi). Then, forϕ ∈ L1(Tm), we have‖ϕ‖BMO(Tm)  2(2/d({δi}mi=0))m max0im{∥∥ϕ(· − i δ2π)∥∥BMOD(Tm)}.Remark 7. To extend our results toRm, denote byD(R) the family of the usual dyadicσ -algebra onR. For0 < δ < 1 with d(δ) > 0, choose an increasing family of dyadicσ -algebraDδ(R) = (Dδn)n∈Z(R) such that, forneven,Dδn(R) = σ({Dδ,kn}k∈Z),Dδ,kn (R) ={(k2n + δ, k+12n + δ], n  0,(k2n + δ +∑0j=n+2 12j ,k+12n + δ +∑0j=n+2 12j], n < 0.Note that allDδn(R)’s are given after fixingDδn(R)’s for all evenn’s. Let {δi}mi=0 be a sequence in(0,1) such thatd({δi}mi=0) > 0. Let iDδ(Rm) = {iDδn(Rm)}n∈Z, whereiDδn(Rm) is them times product of theσ -algebraDδin (R).Then, by the same idea as above, we can get‖ϕ‖BMO(Rm)  max0im‖ϕ‖BMOiDδ (Rm) ∀ϕ ∈ L1(Rm).AcknowledgementThe author is very grateful to Q. Xu and his adviser G. Pisier for useful conversations.References[1] J.B. Garnett, P.W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (2) (1982) 351–371.[2] J.B. Garnett, Bounded Analytic Functions, in: Pure Appl. Math., Vol. 96, Academic Press, New York, 1981.[3] T. Mei, Operator valued Hardy spaces, Preprint.[4] S. Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operator with matrix symbol, C. R. Acad. Sci. Paris, Ser. I 33455–460.

BMO is the intersection of two translates of dyadic BMO

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