Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces

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Journal of Mathematical Analysis and Applications

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Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces

Zoe Nieraeth^a, Guillermo Rey^b,∗

a BCAM–BasqueCenterforAppliedMathematics,Bilbao,Spain

bUniversidadAutónomadeMadrid,Spain

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received5July2022

Availableonline16December2022 Submittedby M.J.Carro

Keywords:

Sparseoperators Muckenhouptweights Banachfunctionspaces Calderón-Zygmundoperators BMO

RubiodeFranciaextrapolation

InthispaperweprovesharpweightedBMOestimatesforsingularintegrals,and weshowhowsuchestimatescanbeextrapolatedtoBanachfunctionspaces.

©2022TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticle undertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

In the 70’s Muckenhoupt and Wheeden extended the known L^∞ → BMO estimates for the Hilbert transformtotheweightedsetting.Inparticular,theyprovedthat



I

|Hf − HfI| dx wfwL^∞



I

w⁻¹dx,

whereHfI istheaverageofHf overI,holdsifandonlyifw⁻¹∈ A∞∩ B2.This isTheorem1in[12].

TheclassB2,whichwasintroducedin[4],is thecollectionofalllocallyintegrableweights w satisfying



I^c

|I|w(x)

|x − cI|²dx wI.

* Correspondingauthor.

E-mailaddresses:[email protected](Z. Nieraeth),[email protected](G. Rey).

https://doi.org/10.1016/j.jmaa.2022.126942

0022-247X/©2022TheAuthor(s). PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

In the same paper (Theorem 2), Muckenhoupt and Wheeden characterize w⁻¹ ∈ A1 as the class of weights w forwhichthefollowingweightedBMOestimateholds

wL^∞(I)

1

|I|



I

|Hf − HfI| dx wfwL^∞ (1.1)

forallintervalsI andallfunctionsf .

This resultwaslaterusedbyHarboure,Macías,andSegoviain[3] toproveaBMO-to-L^p extrapolation result, saying that if a sublinear operator T satisfies the same bound as in (1.1) with implicit constant depending onlyon[w⁻¹]A1,then

T fL^p(v)vfL^p(v)

forallweights v∈ Ap and1< p<∞.

Weshouldmentiontheresultof[14],whereamultilinearextrapolationresultisobtainedwhichgeneralizes and sharpens theonebyHarboure,Macías,and Segovia.Seealso [2], where sharplinearversionsof these two resultsareobtained.Wealsoreferthereaderto[1].

The purpose of this article is twofold: first, we extend the extrapolation result of Harboure, Macías, and Segovia, to the setting of Banach function spaces. Second, we extend the results of Muckenhoupt and Wheeden to sparse operators as well as to Calderón-Zygmundoperators whosekernels satisfy aDini smoothnesscondition.Wefullyrecovertheresultsof[2],andinfactourextrapolationresultissharper,see Remark3.3.

Letus firststateasimplifiedversionofourextrapolationtheorem.

TheoremA.LetT beasublinearoperator. Supposethereisanincreasingfunctionφ: [1,∞)→ (0,∞) such that forallweights w withw⁻¹∈ A1,andallcompactlysupportedfunctionsf with f w∈ L^∞(R^d) wehave

wL^∞(Q)

1

|Q|



Q

|T f − T fQ| dx ≤ φ([w⁻¹]A1)fwL^∞(R^d).

Then, forallBanachfunctionspacesX overR^d forwhichM is boundedonX anditsassociatespace X^, and allcompactly supportedfunctionsf ∈ X

T fX dMX^→X^φ(2MX→X)fX. Ourfull resultisTheorem 3.1insection 3.

As forourresultforCalderón-Zygmundoperators,supposeT isanoperatorrepresentedby

T f (x) =



R^d

K(x, y)f (y) dy

forallx outside ofthesupportoff ,andwhere K satisfies

|K(x, y) − K(z, y)| ≤ Ω

|x − z|

|x − y|

 1

|x − y|^d

for allx,y,z satisfying|x− y|> 2|x− z|> 0,and where Ω: [0,∞)→ [0,∞) is anincreasing subadditive functionwithΩ(0)= 0.

Similartothesituationin[12],itisnotsufficientthatw∈ A∞for T tohaveweightedBMOestimates.

What weneed isageneralization ofthe B₂ conditionthattakes into account theinteraction between the weightandthesmoothnessof thekernel. Thisisquantifiedby

[w]_B(Ω)= sup

Q

|Q|

w(Q)



Q^c

w(x)

|x − cQ|^dΩ (Q)

|x − cQ|

 dx.

Notethat,indimensiononeandwhenΩ(t)= t, [w]_B(Ω)<∞ istheB2 condition.

Withthisdefinition,wehave

TheoremB.Letw⁻¹ ∈ L¹loc(R^d) andlet T bean operatorasabove,then 1

|Q|



Q

|T f − T fQ| dx 

ΩDini+ [w⁻¹]_B(Ω)

[w⁻¹]A_∞w⁻¹QfwL^∞(R^d)

and

wL^∞(Q)

1

|Q|



Q

|T f − T fQ| dx  [w⁻¹]²_A1[w⁻¹]A_∞ΩDinifwL^∞(R^d)

forallcompactlysupportedfunctionsf with f w∈ L^∞(R^d).

This is Theorem 4.4 in section 4. These estimatesare based on sparse domination techniques, so they offerasimplifiedargumenttotheresultsfrom[12] and[13].

Notation.We work in the setting of R^d equipped with the Lebesgue measure dx. For a measurable set E⊆ R^d wedenoteitsmeasure by|E|.Forameasurablefunctionf ∈ L⁰(R^d),ameasurableset E⊆ R^d of finitepositive measure,andaq∈ (0,∞) wewrite

fq,E :=

⎛

⎝ 1|E|



E

|f|^qdx

⎞

⎠

1 q

,

anddenote theessentialsupremum off inE byf∞,E.Moreover, wedenotethelinearizedmeanoff on E by

fE:= 1

|E|



E

f dx.

Byacube inR^d we meanahalf-opencubewhose sidesareparalleltothecoordinateaxes.

WewillwriteAa,b,...B tomeanthatthereisaconstantC dependingonlyontheparametersa,b,· · · , suchthatA≤ CB.WewriteAa,b,···B whenbothAa,b,···B andBa,b,···A.

2. Preliminaries 2.1. Dyadic analysis

InthispaperwewillbeworkinginR^d,butourresultsarealsovalidinmoregeneralspacesofhomogeneous type. We will often reduceour arguments to dyadicgrids: a dyadicgrid is acollection of cubes with the property

P∩ Q = ∅ =⇒ P ⊆ Q or Q ⊆ P.

For acollectionofcubes _E inadyadicgrid_D andacube Q₀∈D,we let_E(Q₀) denote thecubesin_E thatarecontainedinQ₀.Werefertheinterestedreaderto themonograph[11].

2.2. Muckenhouptweights

A weightw in R^d can be associated with the measure through w(E):=

Ew dx. A classicalresult by Muckenhoupt isthattheHardy-Littlewoodmaximaloperator

M f := sup

Q f1,Q1Q,

where the supremum is takenover allcubes Q in R^d, is bounded L^p(R^d,w)→ L^p(R^d,w) for p∈ (1,∞) precisely whenw satisfiestheA_pcondition

[w]Ap:= sup

Q w1,Qw^−1p^p_p,Q<∞,

and boundedL¹(R^d,w)→ L^1,∞(R^d,w) ifandonlyifw satisfiestheA₁condition [w]_A1:= sup

Q w1,Qw⁻¹∞,Q=(Mw)w⁻¹L^∞(R^d)<∞.

However, in thecase p= ∞, we haveL^∞(R^d,w) = L^∞(R^d) forall weights w and hence, M is bounded L^∞(R^d,w)→ L^∞(R^d,w) forallweights w.

Instead, ifwe defineL^p_w(R^d) throughfL^pw(R^d):=fwL^p(R^d), then wedoget ameaningfulcondition for p=∞. Notethatforp∈ (1,∞) we haveL^p_w(R^d)= L^p(R^d : w^p) andhence,M : L^p_w(R^d)→ L^pw(R^d) is bounded preciselywhen

[w^p]

1 p

Ap= sup

Q wp,Qw⁻¹p^,Q<∞.

Forp=∞,thisexpression yieldsthecondition sup

Q w∞,Qw⁻¹1,Q=M(w⁻¹)wL^∞(R^d)= [w⁻¹]_A1.

Indeed,definingL^∞_w(R^d) throughfL^∞_w(R^d)=fwL^∞(R^d),wehavethefollowingresult:

Proposition 2.1.Letw beaweightinR^d.ThenM isboundedL^∞_w(R^d)→ L^∞w(R^d) ifandonlyif w⁻¹∈ A1. In this casewehave

ML^∞_w(R^d)→L^∞w(R^d)= [w⁻¹]A1.

Proof. If M isboundedL^∞_w(R^d)→ L^∞w(R^d),thensincef = w⁻¹ satisfiesfL^∞_w(R^d)= 1,we have [w⁻¹]A1 =M(w⁻¹)L^∞_w(R^d)≤ ML^∞_w(R^d)→L^∞w(R^d).

Fortheconverse,ifw⁻¹∈ A1, thenforallf ∈ L^∞w(R^d) wehave

MfL^∞_w(R^d)≤ fL^∞_w(R^d)M(w⁻¹)L^∞_w(R^d)= [w⁻¹]_A1fL^∞_w(R^d). Theassertionfollows. _

If_D isadyadicgridinR^d,thenwedefine

M^Df := sup

Q∈Df1,Q1Q

andtheclassA1(D) through

[w]A1(D):= sup

Q∈Dw1,Qw⁻¹∞,Q=(M^Dw)w⁻¹L^∞(R^d). Then,exactlyasintheaboveresult,wehaveM^DL^∞_w(R^d)→L^∞w(R^d)= [w⁻¹]_A1(D).

For acube Q∈ D we denote by D(Q) the collectionof cubes inD thatare contained in Q. Defining M^D(Q) accordingly,wedefine theFujii-WilsonA_∞ constantofaweightthrough

[w]A_∞(D):= sup

Q∈D

1 w(Q)



Q

M^D(Q)w dx.

Finitenessofthisconstantcharacterizestheclass

p≥1Ap(D).Notethatinparticularwehave [w]_A∞(D)≤ [w]A1(D).

Inthenon-dyadiccase,theFujii-Wilsoncharacteristic isdefinedas [w]A_∞ := sup

Q

1 w(Q)



Q

M (1Qw) dx,

wherethesupremumistakenoverallcubes.

Wehavethefollowing well-knownresult(which,infact,is acharacterizationofA_∞(D)):

Proposition2.2. LetS ⊆D beanη-sparse collectionofcubesandletQ₀∈D.Thenforaweightw wehave



Q⊆QQ∈S0

w(Q)≤ η⁻¹[w]_A∞(D)w(Q0).

Proof. Wehave



QQ∈S⊆Q0

w(Q)≤ η⁻¹ 

QQ∈S⊆Q0

y∈Qinf M^D(Q0)w(y)|EQ| ≤ η⁻¹

Q0

M^D(Q0)w dx≤ η⁻¹[w]_A∞(D)w(Q0),

asdesired. 

2.3. WeightedBMOspaces

ThespaceBMO(R^d) isdefinedasthespaceoffunctionf ∈ L⁰(R^d) forwhichthesharpmaximalfunction M^f := sup

Q f − fQ1,Q1Q

liesinL^∞(R^d).Thus,itissensibletodefineaweightedanalogueBMO_w(R^d) asthosef ∈ L⁰(R^d) forwhich M^f ∈ L^∞w(R^d).This isfacilitatedthroughthefollowingdefinition.

Definition 2.3.Foraweightw wedefine thespacesBMO¹_w(R^d) andBMO^∞_w(R^d) through

fBMO¹_w(R^d):= sup

Q

f − fQ1,Q

w⁻¹1,Q

= sup

Q

1 w⁻¹(Q)



Q

|f − fQ| dx,

fBMO^∞_w(R^d):= sup

Q w∞,Qf − fQ1,Q, where functionsareidentifiedmoduloconstants.

Wealsodefine theweakanaloguesBMO^1,weak_w (R^d),BMO^∞,weak_w (R^d) ofthese spacesthrough

fBMO^1,weak_w (R^d):= sup

Q c∈Rinf

(f − c)¹QL^1,∞(Q)

w⁻¹(Q) ,

fBMO^∞,weak_w (R^d):= sup

Q

c∈Rinfw∞,Q|Q|⁻¹(f − c)¹QL^1,∞(Q), where functionsareidentifiedmoduloconstants.

For adyadicgrid D inR^d,thespacesBMO¹_w(D),BMO^∞_w(D),BMO^1,weak_w (D),and BMO^∞,weak_w (D) are definedanalogously,thistimetakingthesupremumoverallcubesin_D,andfunctionsareidentifiedmodulo constants oncubesin_D.

Notingthat

w∞,Q

|Q| = 1

(ess infQw⁻¹)|Q| ≥ 1 w⁻¹(Q) forallcubes Q inR^d, wehave

fBMO¹_w(R^d)≤ fBMO^∞_w(R^d), fBMO^1,weak_w (R^d)≤ fBMO^∞,weak_w (R^d)

and hence,BMO^∞_w(R^d) ⊆ BMO¹w(R^d) and BMO^∞,weak_w (R^d) ⊆ BMO^1,weakw (R^d). Conversely, ifw⁻¹ ∈ A1, we have

w∞,Q≤ [w⁻¹]_A1

w⁻¹1,Q

forallcubes Q inR^d. Thus,inthiscasewehave

fBMO^∞_w(R^d)≤ [w⁻¹]A1fBMO¹_w(R^d), (2.1)

fBMO^∞,weak_w (R^d)≤ [w⁻¹]A1fBMO^1,weak_w (R^d), (2.2) and BMO^∞_w(R^d)= BMO¹_w(R^d),BMO^∞,weak_w (R^d)= BMO^1,weak_w (R^d).

Wealsonote thatL^∞_w(R^d)⊆ BMO¹w(R^d) with

fBMO¹_w(R^d)≤ 2fL^∞_w(R^d). Thus, ifw⁻¹∈ A1,wehaveL^∞_w(R^d)⊆ BMO^∞w(R^d) with

fBMO^∞_w(R^d)≤ 2[w⁻¹]A1fL^∞_w(R^d).

The followingresult showsthatthespace BMO^∞_w(R^d) consists ofprecisely those f ∈ L⁰(R^d) for which M^f ∈ L^∞w(R^d).

Proposition2.4. Wehave

fBMO^∞_w(R^d)=M^fL^∞_w(R^d), fBMO^∞,weak_w (R^d)=Mweak^ fL^∞_w(R^d), where

M_weak^ f := sup

Q

cinf∈R|Q|⁻¹(f − c)¹QL^1,∞(Q)1Q.

Foradyadicgrid _D inR^d,analogously definingthesharp maximaloperators M^,D andM_1,^,_∞^D with respect tothis grid,we alsohave

fBMO^∞_w(D)=M^,DfL^∞_w(R^d), fBMO^1,∞_w (D)=Mweak^,D fL^∞_w(D).

Proof. We onlyprovethefirstequality, theothersbeing analogous. Since 1Qw≤ w∞,Q for allcubesQ, wehave

(M^f )w = sup

Q f − fQ1,Q1_Qw≤ fBMO^∞_w(R^d).

Thisproves theinequality(M^f )wL^∞(R^d)≤ fBMO^∞_w(R^d).Fortheconverse,fixacubeQ andletε> 0.

Then theset of x∈ Q such thatw∞,Q≤ (1+ ε)w(x) has positive measure.Hence, there arex∈ Q for which

w∞,Qf − fQ1,Q≤ (1 + ε)f − fQ1,Qw(x)1_Q(x)≤ (1 + ε)(M^f )wL^∞(R^d)

Thus,fBMO^∞_w(R^d)≤ (1+ ε)(M^f )wL^∞(R^d).Lettingε→ 0,theassertionfollows.  Proposition2.5. Letw be aweight.Wehave

fBMO¹_w(R^d) sup

Q c∈Rinf

f − c1,Q

w⁻¹1,Q

,

fBMO^∞_w(R^d) sup

Q

c∈Rinfw∞,Qf − c1,Q.

Thus, inparticular,

fBMO^1,weak_w (R^d) fBMO¹_w(R^d),

fBMO^∞,weak_w (R^d) fBMO^∞_w(R^d). This similarlyholds forBMO¹_w(D), BMO^∞_w(D) foradyadicgrid D in R^d.

Proof. Weonlyprovethesecond equivalence,thefirstonebeing analogous.Theinequality≥ followsfrom takingc=fQ.Fortheconverseinequality,fixacubeQ andnotethatforanyc∈ R wehave

f − fQ1,Q≤ f − c1,Q+|f − cQ| ≤ 2f − c1,Q.

Thus,fBMOw(R^d)≤ 2sup_Qinf_c∈Rw∞,Qf − c1,Q.AstheresultfollowsanalogouslyforBMO_w(_D),this provestheassertion. _

Intheunweightedcase,itfollowsfromtheJohn-StrömbergcharacterizationofBMO thattheweakBMO spaceisequivalenttotheusualone.Moreprecisely,denotingthenon-decreasingrearrangementofafunction f byf^∗,foracubeQ andλ∈ (0,1) we define

M_λ^f := sup

Q

ω_λ(f ; Q)1_Q,

where

ωλ(f ; Q) = inf

c∈Rinf{t > 0 : |{x ∈ Q : |f − c| > t}| ≤ λ|Q|}.

Observethat

ωλ(f ; Q)≤ λ⁻¹|Q|⁻¹ inf

c∈Rf − cL^1,∞(Q), so thatM_λ^f ≤ λ⁻¹M_weak^ f .

We canextend John-Strömberg’s characterization of BMOthrough medians (see [15]) to theweighted setting usingthefollowingpointwise versionbyA. Lerner[7,Theorem 1.3]:

M^f d M (M^1

2f ), (2.3)

forallf ∈ L¹loc(R^d).

Proposition 2.6. Letw⁻¹∈ A1.ThenBMO^∞_w(R^d)= BMO^∞,weak_w (R^d) with

fBMO^∞_w(R^d)d[w⁻¹]_A1fBMO^∞,weak_w (R^d). Proof. ByProposition2.4,(2.3),and Proposition2.1, wehave

fBMO^∞_w(R^d)dM(M^1

2

f )L^∞_w(R^d)≤ [w⁻¹]A1M^1

2

fL^∞_w(R^d). Theresultnowfollows fromthefactthatM^1

2

f  Mweak^ f . _ 2.4. Banachfunctionspaces

Wedenote thepositive measurablefunctionsonR^d byL⁰(R^d)+. Definition 2.7.Supposeρ: L⁰(R^d)₊→ [0,∞] satisfies

(i) ρ(f )= 0 ifandonlyiff = 0 a.e.;

(ii) ρ(f + g)≤ ρ(f)+ ρ(g) and ρ(λf )= λρ(f ) forallf,g∈ L⁰(R^d)+ and scalarsλ≥ 0;

(iii) foreverysequence(fn)_n∈N inL⁰(R^d)+ satisfying0≤ fn ↑ f pointwisea.e.forf ∈ L⁰(R^d)+,wehave ρ(fn)↑ ρ(f).

(iv) foreverymeasurableE⊆ R^d ofpositivemeasurethereexistsameasurableF ⊆ E ofpositivemeasure withρ(1_F)<∞.

Then wecallρ afunctionnorm. Moreover,wecallthespaceX ⊆ L⁰(Rⁿ) off ∈ L⁰(R^d) forwhich

fX := ρ(|f|) < ∞,

aBanachfunctionspaceoverR^d.

Givenafunctionnormρ,wedefineanewfunctionnorm ρ^ through ρ^(g) := sup

ρ(f )=1fgL¹(R^d).

TheKöthedual ofX isdefinedas theBanachfunctionspaceassociatedto ρ^,i.e., X^ :={g ∈ L⁰(R^d) : f g∈ L¹(R^d) for all f ∈ X}

with

gX^ = sup

f X=1fgL¹(R^d).

Property (iv) is called thesaturation property of ρ and is equivalent to the existence of a weakorder unit,i.e.,anf > 0 forwhichρ(f )<∞.Property (iv)isalsoequivalentto thefactthatρ^ satisfies(i)and hence,isafunctionnorm. Thus,X^ isaBanachfunctionspace overR^d.

The Fatou property (iii) is equivalent to the assertion that ρ^ = ρ and hence, X^ = X isometrically.

Thus,inparticularwehave

fX = sup

g X=1fgL¹(R^d).

We say that a Banachfunction space X over R^d is order-continuous when for all (f_n)_n∈N inX with f_n ↓ 0 a.e. wehavefnX → 0. Wenote thatX is reflexiveifandonlyifX and X^ areorder-continuous.

Forproofsoftheaboveclaimswe referthereaderto[16].

Bycarefullytracking theconstantsintheproofof[10,Corollary 4.3],wearriveat thefollowing quanti- tativeversion:

Theorem 2.8.Let X be aBanach function spaceover R^d andsuppose that M : X → X is bounded. Then thefollowingassertionsare equivalent:

(i) There isaconstant C_X> 0 such that

fX ≤ CXM^fX

forallf ∈ X withtheproperty that |{x∈ R^d:|f(x)|> α}|<∞ forallα > 0;

(ii) There isaconstant C_X,weak> 0 such that

fX ≤ CX,weakMweak^ fX

forallf ∈ X withtheproperty that |{x∈ R^d:|f(x)|> α}|<∞ forallα > 0;

(iii) M : X^→ X^ isbounded.

Moreover, ifC_X andC_X,weak are thesmallest possibleconstant in(i) and(ii),then

CX≤ CX,weakdMX^→X^ dCXMX→X. (2.4) Proof. Ourstrategywill betoprovethat(ii)⇒(i)⇒(iii)⇒(ii).

Theassertion(ii)⇒(i)withC_X ≤ CX,weak followsfromthefactthatM_weak^ f ≤ M^f .

For(i)⇒(iii)weusethatitwasshownin[10, Theorem1.1] that(i)isequivalenttotheassertion

(iv) thereisaC > 0 suchthat

fMgL¹(R^d)≤ CMfXgX^

forallf ∈ L¹loc(R^d) andg∈ X^.

Moreover, ifC isthesmallestpossibleconstantinthisestimate, then CdCX.

Thus, wehave

MgX^ = sup

f X=1fMgL¹(R^d)≤ C sup

f X=1MfXgX^ = CMX→XgX^

so thatM : X^→ X^ withMX^→X^ d CXMX→X,asdesired

For (iii)⇒(ii), we use [8, Theorem 1] which states thatthere is adimensional constantλ∈ (0,1) such thatforallf satisfyingthepropertythat|{x∈ R^d:|f(x)|> α}|<∞ forallα > 0 andallg∈ L¹loc(R^d) we have

fgL¹(R^d)d (Mλ^f )(M g)L¹(R^d). Since M_λ^f ≤ λ⁻¹M_weak^ f ,itfollowsthat

fX = sup

g X=1fgL¹(R^d)d sup

g X=1(Mweak^ f )(M g)L¹(R^d)

≤ sup

g X=1Mweak^ fXMgX^ =MX^→X^Mweak^ fX. This proves(ii)withC_X,weakdMX^→X^ and (2.4).Theassertionfollows. _ 3. RubiodeFrancia extrapolationfromweightedBMO

Theorem 3.1. LetT be anoperatorin L⁰(R^d). Supposethere isan increasing functionφ: [1,∞)→ (0,∞) such thatforallweights w⁻¹∈ A1 andallcompactlysupportedfunctionsf ∈ L^∞w(R^d) wehave

T fBMO^∞_w(R^d)≤ φ([w⁻¹]A1)fL^∞_w(R^d). (3.1) Then, forallBanach functionspacesX over R^d forwhichM is boundedon X andX^,andallcompactly supportedfunctions f ∈ X wehave

T fX≤ 2CXφ(2MX→X)fX,

where C_X istheconstantfromTheorem2.8.IfT isa(sub)linear operatorandX is order-continuous,then T extends toaboundedoperatorX → X satisfying

T X→X ≤ 2CXφ(2MX→X), (3.2)

in particular

T X→XdMX^→X^φ(2MX→X). (3.3) An analogousassertionholds whenyou replaceBMO^∞_w(R^d) byBMO^∞,weak_w (R^d) andC_X byC_X,weak.

WeneedthefollowinglemmawhichisbasedontheRubiodeFranciaalgorithm.

Lemma3.2. LetX beaBanach functionspaceoverR^d forwhichM : X→ X. Thenforallf ∈ X, g∈ X^ thereexists aweightw⁻¹∈ A1 suchthat

[w⁻¹]_A1 ≤ 2MX→X, (3.4)

andf ∈ L^∞w(R^d),g∈ L¹_w−1(R^d) with

fL^∞_w(R^d)gL¹_w−1(R^d)≤ 2fXgX^. (3.5) Proof. Weset

w⁻¹:=

∞ k=0

M^kf 2^kM^k_X→X,

wherewe haverecursivelydefinedM⁰f :=|f|,M^k+1f := M (M^kf ).Then wehave

M (w⁻¹)≤

∞ k=0

M^k+1f

2^kM^k_X→X ≤ 2MX→Xw⁻¹, proving(3.4).

Since|f|= M⁰f ≤ w⁻¹,wehavefL^∞_w(R^d)≤ 1. Combiningthis withthefactthatw⁻¹X ≤ 2fX, wehave

fL^∞_w(R^d)gL¹_w−1(R^d)≤ gw⁻¹L¹(R^d)≤ w⁻¹XgX^ ≤ 2fXgX^.

Thisproves (3.5),asdesired. 

Proof of Theorem3.1. Letf ∈ X havecompactsupportandletg∈ X^ withgX^ = 1.ByLemma3.2we canpickaweightw⁻¹ ∈ A1 suchthat[w⁻¹]A1 ≤ 2MX→X andf ∈ L^∞w(R^d),g∈ L¹_w−1(R^d) with

fL^∞_w(R^d)gL¹

w−1(R^d)≤ 2fX. Hence,byProposition2.4 wehave

M^(T f )gL¹(R^d)≤ M^(T f )L^∞_w(R^d)gL¹_w−1(R^d)≤ φ([w⁻¹]A1)fL^∞_w(R^d)gL¹_w−1(R^d)

≤ 2φ(2MX→X)fX.

Hence, since f has compact supportand thus |{|f| > λ}|≤ |supp f| < ∞ for all λ> 0, it follows from Theorem2.8,that

T fX ≤ CXM^(T f )X= C_X sup

g X=1M^(T f )gL¹(R^d)

≤ 2CXφ(2MX→X)fX, provingthefirstresult.

Forthenextassertion,sinceX isordercontinuousthefunctionsinX ofcompactsupportaredenseinX.

Indeed,givenf ∈ X,setf_n := f1B(0;n).Thenf_n∈ X hascompactsupportand|f −fn|=|f|¹R^d\B(0;n)↓ 0.

Hence,f − fnX → 0,asdesired. Thus,theresultfollowsfrom thefactthatifT is(sub)linear, thenitis uniformly continuousandhence,extendstoallofX withthesamebound.

TheassertionforBMO^∞_w(R^d) replacedbyBMO^∞,weak_w (R^d) andCX replacedbyCX,weak isprovedanal- ogously. 

Remark3.3. Itwas shownin[2] that theinitial estimate(3.1) implies thatforp∈ (1,∞) and w∈ Ap we have

T L^p(w)dML^p(R^d;w^1−p)φ(ML^p(R^d;w)).

We recoverthis resultin(3.3),sinceifX = L^p(R^d;w),then X^ = L^p(R^d;w^1−p). Asamatterof fact,we improve onthis estimatein(3.2).Indeed,forX = L^p(w;R^d) weobtain

T L^p(w)≤ 2CXφ(ML^p(R^d;w)), where C_X istheconstantappearingintheFefferman-Stein inequality

fL^p(R^d;w)≤ CXM^fL^p(R^d;w).

While ML^p(R^d;w^1−p) isbounded if andonly ifw ∈ Ap, theconstantCX is bounded undermuchmore generalconditions.Forexample,itisboundedwhen w∈ A∞.

4. WeightedBMOestimatesforsingularintegrals

We will begin this section discussing the action of sparse operators on L^∞_w. These serve as a way to gainintuitionforthemorecomplicatedCalderón-Zygmundoperators.Wenote,however,thatunlikeinthe classical L^p case, estimatesfor sparse operators donotimmediately imply bounds forCalderón-Zygmund operators becauseBMOis notalattice.

4.1. Sparseoperators

For η ∈ (0,1),acollectionof setsS in R^d iscalled η-sparse ifthere exists apairwisedisjoint collection ofsets{ES}S∈S suchthatES ⊆ S and|ES|≥ η|S| forallS∈ S.Forsuchacollectionwedefinethesparse operator associated toS as

ASf := 

Q∈S

f1,Q1Q.

Ingeneral,theseoperatorsdonot mapL^∞toBM O.Thereareseveralobstaclesthatpreventthis,which we nowdescribe.

Forany familyS ofsetsinR^d,leth_S be itsheightfunction:

h_S =

J∈S

1J.

FirstwenotethatASf maynotevenbelocallyintegrableforf ∈ L^∞(R^d).Infact,thesituationismuch worse.Indeed, consider thesetsFn = [0,1)∪ [n,n+ 1) forn≥ 1 and define thefamilyS = {Fn : n≥ 1}.

Foreachn≥ 1 thesubsetE_n= [n,n+ 1)⊆ Fn hasmeasure1= ¹₂|Fn| and thecollection{En} ispairwise disjoint,thusS is ¹₂-sparse.However

AS(1_[0,1)) =

∞ n=1

1

2^1Fn= 1 2h_S isidentically∞ on[0,1).

Ingeneral,weneedsome boundednessofthemaximalfunctionassociatedtoS forAS tobehavewell.In particular,ifS consists ofintervals(orgenerallycubesinanydimension),thenAS isboundedonL²(R^d), so we have no problems with the local integrability of AS(f ).But even this is not enough to reasonably defineAS as anoperatorfromL^∞(R^d) toBMO(R^d).

Toseethis,considerthefamilyS = {[0,2ⁿ): n≥ 0}.Thisfamilyis ¹₂-sparse,howeverifwedefineAS(1) withthenaturalpointwiselimitweobtainAS(1)=∞ on[0,∞).

Ingeneral,weneedS tohavefinitetotalmeasure,atleastqualitatively.However,thisisalsonotenough because there may still be issues similar to those that appear when comparing BMO and dyadic BMO.

Indeed,consider

S = {[0, 2⁻ⁿ) : n≥ 0}.

Thisfamilyis ¹₂-sparseandhasfinitetotalmeasure,thusAS(f ) isintegrableforallf ∈ L^∞(R^d).However, lettingI_n= [0,2⁻ⁿ) andJ_n= [−2⁻ⁿ,2⁻ⁿ) forn≥ 1,wehave

1

|Jn|



Jn

AS(1_[0,1)) dx = 2ⁿ⁻¹

∞ m=0

|Im∩ In|

= 2ⁿ⁻¹

n m=0

|In| + 2ⁿ⁻¹

∞ m=n+1

|Im|

= 2ⁿ⁻¹(n + 1)2⁻ⁿ+ 2ⁿ⁻¹

∞ m=n+1

2^−m

= n 2 + 1.

Thus

1

|Jn|



Jn

|AS(1_[0,1))− AS(1_[0,1))Jn| dx ≥ 1

|Jn|

0

−2⁻ⁿ

n 2 + 1

dx

≥ n 4 andhence,AS(1_[0,1)) isnotinBMO(R^d).

Whenthesparsecollectionconsistsofcubesbelongingtothesamedyadicfamily_D,thenwecan obtain BMO(D) estimates.

Theorem4.1. Letw⁻¹∈ L¹loc(R^d) andletS ⊆D bean η-sparsecollection of cubes.Then

ASfBMO¹_w(D) η⁻¹[w⁻¹]A_∞(D)fL^∞_w(R^d), (4.1)

ASfBMO^∞_w(D) η⁻¹[w⁻¹]_A1(D)[w⁻¹]_A∞(D)fL^∞_w(R^d) (4.2) and