Boundedness of singular integrals on C1,α intrinsic graphs in the Heisenberg group

Boundedness

of

singular

integrals

on

C

intrinsic

graphs

in

the

Heisenberg

group

Vasileios Chousionisa_{, Katrin Fässler}b,∗_{, Tuomas Orponen}c

a _{Departmentof Mathematics,UniversityofConnecticut,USA} b_{DepartmentofMathematics,UniversityofFribourg,Switzerland}

c_{DepartmentofMathematicsandStatistics,Universityof Helsinki,Finland}

a b s t r a c t MSC: primary42B20 secondary31C05,35R03,32U30, 28A78 Keywords: Singularintegrals Heisenberggroup

Removablesetsforharmonic functions

We study singular integral operators induced by 3-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is L2_{bounded on vertical planes, with}

uniform constants, then it is also L2_{bounded on all intrinsic}

graphs of compactly supported C1,α _{functions over vertical}

planes.

In particular, the result applies to the operator R induced by the kernel

K(z) = ∇Hz−2, z ∈ H \ {0},

the horizontal gradient of the fundamental solution of the sub-Laplacian. The L2 _{boundedness of R is connected with the}

question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that the intrinsic graphs mentioned above are non-removable. Apart from subsets of vertical planes, these are the first known examples of

non-✩ _{V.C.issupportedbytheSimonsFoundationviatheproject‘AnalysisanddynamicsinCarnotgroups’,}

Collaborationgrantno. 521845.K.F.is supportedbySwissNationalScienceFoundationviatheproject ‘IntrinsicrectifiabilityandmappingtheoryontheHeisenberggroup’,grantno.161299.T.O.issupported bytheAcademyofFinlandviatheproject‘Restrictedfamiliesofprojectionsandconnections toKakeya typeproblems’,grantno.274512.

* Correspondingauthor.

E-mailaddresses:[email protected](V. Chousionis), katrin.s.fassler@jyu.fi(K. Fässler),

tuomas.orponen@helsinki.fi(T. Orponen).

http://doc.rero.ch

Published in "Advances in Mathematics 354(): 106745, 2019"

which should be cited to refer to this work.

removable sets with positive and locally finite 3-dimensional measure.

Contents

1. Introduction . . . 2

2. Definitions and preliminaries . . . 5

2.1. The Heisenberg group and general notation . . . 5

2.2. Kernels and singular integral operators in H . . . 6

2.3. Intrinsic graphs and the boundedness of SIOs . . . 11

3. Proof of the main theorem . . . 17

3.1. Christ cubes and the T 1 theorem . . . 18

3.2. Verifying the T 1 testing condition . . . 19

4. Hölder regularity and linear approximation . . . 28

5. Boundedness of the Riesz transform, and removability . . . 36

References . . . 43

1. Introduction

Thepurposeofthispaperistostudytheboundednessofcertain3-dimensional singu-larintegralsonintrinsicgraphsinthefirstHeisenberggroupH,a3-dimensionalmanifold witha4-dimensionalmetricstructure.Alltheformaldefinitionswillbe deferredto Sec-tion2,so thisintroductionwillbe brief,informalandnotentirelyrigorous.

Westudysingular integraloperators (SIOs)of convolutiontype.InH,this refersto objectsofthefollowingform:

Tμf (p) =



K(q−1· p)f(q) dμ(q), (1.1)

whereK :H \ {0}→ Rd_{isakernel,andμ isalocallyfiniteBorelmeasure.Specifically,}

weare interestedintheL2 _{boundedness oftheoperatorf → T}

Hf on certain3-regular

surfaces Γ ⊂ H, where H = H3_|

Γ is 3-dimensional Hausdorff measure restricted to

Γ. The relevant surfaces Γ are the intrinsic Lipschitz graphs, introduced by Franchi, Serapioni and Serra Cassano [24] in 2006. These are the Heisenberg counterparts of (co-dimension1)LipschitzgraphsinRd_{.IntheEuclideanenvironment,theboundedness}

ofSIOsonLipschitzgraphs,andbeyond,isaclassicaltopic,developedbyCalderón[6], Coifman-McIntosh-Meyer[15], David[16],David-Semmes[18],andmanyothers.

IfΓ isaverticalplaneW (aplaneinH containingtheverticalaxis),thenthe bound-ednessofT_H onL2_{(H) isessentiallyaEuclideanproblem.Infact,as longasp,q∈ W ,}

thegroupoperationp·q behaveslikeadditioninR2_{.Also,H = H}3_|

W issimplyaconstant

multipleof2-dimensionalLebesguemeasure.So,THcanbeidentifiedwithaconvolution

typeSIOinR2_.1_TheL2_{boundednessquestionforsuchoperatorsisclassical,seeStein’s}

book[34], andforinstanceFourier-analytictools areapplicable.

The main result of the paper, see Theorem 1.1 below, asserts that solvingthe Eu-clideanproblemautomaticallyyieldsinformationonthenon-Euclideanproblem.Before makingthatstatementmorerigorous,however,weask:whatarethenaturalSIOsinH, inthecontextofthe3-dimensionalsurfacesΓ?InRd_{,aprototypicalsingularintegralis}

the(d− 1)-dimensionalRiesztransform,whosekernelisthegradientofthefundamental solutionoftheLaplacian,

KRd(x) =∇|x|−(d−2).

The boundedness of the associated singular integral operator R_Rd is connected with the problem of removability for Lipschitz harmonic functions. A closed set E ⊂ Rd

is removable for Lipschitz harmonic functions, or just removable, if whenever D ⊃ E is open, and f : D → R is Lipschitz and harmonic in D\ E, then f is harmonic in D. In brief, the connection between R_Rd and removability is the following:if R_Rd is bounded onΓ forsomeclosed (d− 1)-regularset Γ,then Γ,orpositivemeasure closed subsets of Γ, are not removable for Lipschitz harmonic functions, see Theorem 4.4 in [28]. Theimportanceof theRiesz transforminthestudy of removabilityishighlighted intheseminalpapersbyDavidandMattila[17],andNazarov,TolsaandVolberg[30,31]. Using, among other things, techniques fromnon-homogeneous harmonicanalysis, they characteriseremovablesetsasthepurely(d−1)-unrectifiablesetsinRd_{,thatis,thesets}

whichintersecteveryC1_{hypersurfaceinasetofvanishing(d−1)-dimensionalHausdorff}

measure.

InH, thecounterparts ofharmonicfunctionsare solutionstothe sub-Laplace equa-tion Δ_Hu= 0,see Section2.1, or [4]. Withthis notion ofharmonicity, the problem of removabilityinH makes sense,and hasbeenstudiedin[12,10].Also, asinRd_,

remov-ability isconnected with the boundedness ofa certain singularintegralR_H, now with kernel

KH(z) =∇Hz−2,

where·  denotestheKorányidistance.IncontrasttotheEuclideancase,thiskernelis not antisymmetricinthesense K_H(z)=−K_H(z−1). Nevertheless,itis knownthatthe associated SIO R_H is L2 _{bounded onvertical planes, see Remark3.15 in[12]. This is}

duetothefactthatK_Hishorizontally antisymmetric:K_H(x,y,t)=−K_H(−x,−y,t) for (x,y,t)∈ H.On vertical planesW , thisamount ofantisymmetrysuffices toguarantee boundednessonL2 _{byclassicalresults,see forinstanceTheorem4onp.623in[34].}

1 _{Oneshouldkeepinmind,however,thatifK isaCalderón-ZygmundkernelinH,thentherestriction}

ofK toW satisfiesthestandardgrowthandHöldercontinuityestimateswithrespecttoanon-Euclidean metriconW .

Wenowintroducethemaintheorem.WeproposeinConjecture2.11thatconvolution typeSIOswithCalderón-Zygmundkernelswhichareuniformly L2_{-bounded onvertical}

planes,arealsoboundedonintrinsicLipschitzgraphsΓ⊂ H.Thiswouldprovethatsuch sets Γ arenon-removable –afactwhich, before thecurrent paper,was onlyknownfor theverticalplanesW .Inthispaper,weverifyConjecture2.11fortheintrinsicgraphsof compactlysupportedintrinsicallyC1,α_{(W )-functions,definedonverticalplanesW ⊂ H,}

see Definitions 2.12 and 2.16. This class contains all compactly supported Euclidean C1,α_{-functions,withtheidentificationW ∼=R}2_{,see Remark2.21.}

Theorem1.1. Letα > 0,andassumethatφ∈ C1,α_{(W ) hascompactsupport.Then,any}

convolutiontypeSIOwithaCalderón-ZygmundkernelwhichisuniformlyL2_-boundedon

verticalplanes,isboundedonL2_{(μ) forany3-Ahlfors-Davidregularmeasureμ supported}

ontheintrinsicgraphΓ ofφ.Inparticular,this istrueforthe3-dimensionalHausdorff measureonΓ.

Inparticular,theresultappliestotheoperatorR_H,asitsL2_{-boundednessonvertical}

planesisknown.Theformalconnectionbetweentheboundednessofthesingularintegral RH, andremovability,isexplainedinthefollowingresult:

Theorem 1.2. Assume that μ is a non-trivial positive Radon measure on H, satisfying thegrowth condition μ(B(p,r))≤ Cr3 _{forp∈ H and r > 0,and such that thesupport}

spt μ has locally finite 3-dimensional Hausdorff measure. If R_H is bounded on L2_(μ),

thenspt μ isnot removableforLipschitzharmonic functions.

WeproveTheorem1.2inSection5;theargumentisnearlythesameas theoneused byMattilaandParamonov[28] intheEuclideancase.Thereareafewsubtledifferences, however,soweprovideallthedetails.Theproofalsorequiresanauxiliaryresultofsome independentinterest,onslicingaset inH byhorizontallines, seeLemma 5.3.

WithTheorems1.1and1.2inhand,thefollowing corollariesareratherimmediate: Corollary 1.3. Letα > 0. Assume that φ ∈ C1,α_{(W ) is compactlysupported. If E is a}

closed subsetof theintrinsicgraph of φ withpositive3-dimensionalHausdorffmeasure, thenE is notremovable.

Corollary1.4.Letα > 0.Assumethat Ω⊂ R2∼_{=W isopen,andφ isEuclideanC}1,α _on

Ω.If E isaclosedsubsetof theintrinsicgraph ofφ overW withpositive3-dimensional Hausdorffmeasure, thenE isnot removable.

Thestructureofthepaperisthefollowing.InSection2,weintroducealltherelevant concepts, from singular integrals to (intrinsic) C1,α functions, and prove some simple lemmas. InSection 3, we proveTheorem 1.1. InSection 4, we study how well the (in-trinsic) graphs of C1,α _{functions are approximated by vertical planes; this analysis is}

requiredintheproofofTheorem1.1.Finally,inSection5,westudytheconnectionwith theremovabilityproblem,and proveTheorem1.2 andCorollaries1.3and1.4.

2. Definitionsandpreliminaries

2.1. The Heisenberggroupandgeneral notation

ThefirstHeisenberggroupH isR3 endowedwiththegrouplaw

z1· z2= (x1+ x2, y1+ y2, t1+ t2+1₂[x1y2− x2y1]), (2.1)

forzi = (xi,yi,ti)∈ H.Theneutralelementinthesecoordinatesisgivenby0= (0,0,0)

andtheinverseofp= (x,y,t) isdenotedbyp−1andgivenby(−x,−y,−t).TheKorányi distance isdefinedas

d(z1, z2) :=(z2)−1· z1, z1, z2∈ H, (2.2)

where

(x, y, t) :=4 _(x2_{+ y}2₎2_{+ 16t}2_{, for (x, y, t)∈ H}1_{. (2.3)}

Aframe fortheleft invariantvectorfieldsisgivenby

X = ∂x−y2∂t, Y = ∂y+x2∂t, and T = ∂t.

Thehorizontalgradient ofafunctionu: Ω→ R on anopensetΩ⊆ H is ∇Hu = (Xu)X + (Y u)Y ,

andthesub-Laplacian of u is

Δ_Hu = X2u + Y2u. (2.4)

Weconsider the horizontalgradientas amappingwith valuesinR2 _{and write∇} Hu=

(Xu,Y u).Forathorough introductionto theHeisenberg group,wereferthe readerto Chapter 2ofthemonograph[7].

2.1.1. Notation

WeusuallydenotepointsofH byz,p orq;incoordinates,weoftenwritez = (x,y,t) with x,y,t ∈ R. Pointson vertical subgroups W ⊂ H (see Section 2.3.1) are typically denotedbyw.

Unlessotherwisespecified,allmetricconceptsinthepaper,suchasthediameterand distance ofsets,aredefinedusingthemetricd givenin(2.2).Thenotation|· | refersto

Euclideannorm, and·  referstothequantitydefinedin(2.3).Aclosed ballin(H,d) ofradiusr > 0 andcentrez∈ H is denotedbyB(z,r).

For A,B > 0, we use the notation A h B to signify that there exists a constant

C≥ 1,dependingonlyontheparameter“h”,suchthatA≤ CB.Ifno“h”isspecified, the constantC is absolute. We abbreviate the two-sided inequalityA h B h A by

A∼hB.

ThenotationHs_{standsforthes-dimensionalHausdorffmeasure(withrespecttothe}

metricd),and Lebesguemeasure onR2 _{isdenoted byL}2_{; thisnotation isalso usedto}

denoteLebesguemeasure onthesubgroupsW undertheidentificationW ∼=R2. 2.2. Kernels andsingularintegral operatorsinH

An aim of the paper is to study the L2 _{boundedness of singular integraloperators}

(SIOs) on fairly smooth 3-Ahlfors-David regular surfaces in H (see Section 2.3 for a moreprecisedescriptionofoursurfaces).ButwhataretheseSIOs–andwhataretheir kernels?Inthispaper,akernel isanycontinuousfunctionK :H \ {0}→ Rd_.Motivated

bysimilarconsiderationsinEuclideanspaces,itseemsreasonabletoimposethefollowing growthand Höldercontinuityestimates:

|K(z)|  _z13 and |K(z1)− K(z2)| 

z−1

2 · z1β

z13+β

, (2.5)

forsomeβ ∈ (0,1],andforallz∈ H\{0} andz1,z2∈ H\{0} withd(z1,z2)≤ z1/2.We

call such kernels 3-dimensional Calderón-Zygmund (CZ) kernels in H. The conditions above, and the lemma below, imply that our terminology is consistent with standard terminology,seeforinstancep.293inStein’sbook[34].

Lemma2.1. Assume that akernel K :H \ {0}→ Rd _{satisfies thesecond (Hölder}

conti-nuity)estimate in(2.5) forsomeβ > 0. Then, |K(q−1_{· p} 1)− K(q−1· p2)| + |K(p−11 · q) − K(p−12 · q)|  p−1 2 · p1β/2 q−1_{· p1}3+β/2 (2.6) forq∈ H,p1,p2∈ H \ {q} withd(p1,p2)≤ d(p1,q)/2.

Proof. Writez1:= q−1· p1 andz2:= q−1· p2.Thend(z1,z2)≤ z1/2 byleft-invariance

ofd,sothefirstsummandin(2.6) hasthecorrectboundby(2.5),evenwithβ/2 replaced byβ.Hence,tofindaboundforthesecondsummand,weonlyneedtoprovethat

|K(z−1 1 )− K(z2−1)| z −1 2 · z1β/2 z13+β/2 .

Wemaymoreoverassume thatd(z1,z2)≤ z1/C forasuitable largeconstantC≥ 1.

Wewouldliketoapply (2.5) asfollows,

|K(z−1

1 )− K(z2−1)|  z

2· z1−1β

z13+β

, (2.7)

but we first need to make sure thatd(z−11 ,z2−1) ≤ z1/2. Write z1 = (x1,y1,t1) and

z2= (x2,y2,t2),andobservethat

d(z1−1, z2−1) =z2· z1−1 = (x2− x1, y2− y1, t2− t1−12[x2y1− y2x1]  (x2− x1, y2− y1, t2− t1+1₂[x2y1− y2x1] +|x2y1− x1y2| = d(z1, z2) +  |(x2− x1)y1− (y2− y1)x1|  d(z1, z2) +  d(z1, z2)  z1. (2.8)

It follows from (2.8) that d(z−11 ,z2−1) ≤ z1/2, if the constant C was chosen large

enough.Hence,theestimate(2.7) islegitimate,andwemayfurtheruse(2.8) obtain |K(z−1 1 )− K(z−12 )|  z −1 2 · z1β z13+β +z −1 2 · z1β/2z1β/2 z13+β  z−1 2 · z1β/2 z13+β/2 , as claimed. 2

WenowrecallsomebasicnotionsaboutSIOs.Fixa3-dimensionalCalderón-Zygmund kernelK :H \ {0}→ Rd_{,andacomplexRadonmeasureν.For> 0, wedefine}

Tν(p) :=



q−1_·p>

K(q−1· p) dν(q), p∈ H,

whenevertheintegralontherighthandsideisabsolutelyconvergent;thisis,forinstance, the case if ν has finite total variation. Next, fix a positive Radon measure μ on H satisfyingthegrowthcondition

μ(B(p, r))≤ Cr3, p∈ H, r > 0, (2.9)

where C≥ 1 isaconstant.Given acomplexfunctionf ∈ L2_{(μ) and> 0,wedefine}

Tμ,f (p) := T(f dμ)(p), p∈ H.

IteasilyfollowsfromthegrowthconditionsonK andμ,andCauchy-Schwarzinequality, thattheexpressionontherightmakessenseforall> 0.

Definition2.2.Givena3-dimensionalCZkernelK,andameasureμ satisfying(2.9),we saythattheSIOT associatedtoK isboundedon L2_{(μ),iftheoperators}

f → Tμ,f

areboundedonL2_{(μ) withconstantsindependentof> 0.}

For the rest of the paper, we are mainly concerned with measures μ satisfying the 2-sided inequality cr3 _{≤ μ(B(p,r)) ≤ Cr}3 _{for all p ∈ spt μ and 0< r ≤ diam(spt μ),}

andforsomefixedconstants0< c< C <∞.Suchmeasures arecalled3-Ahlfors-David regular,or 3-ADR inshort.

Remark2.3.Given a3-dimensional CZ kernelK, thekernel K∗, definedby K∗(p) := K(p−1),isthekerneloftheformaladjointT_μ,∗ ofTμ, since

 (Tμ,f )g dμ =  ⎛ ⎜ ⎝  q−1_·p> K(q−1· p)f(q) dμ(q) ⎞ ⎟ ⎠ g(p) dμ(p) =  ⎛ ⎜ ⎝  p−1_·q> K∗(p−1· q)g(p) dμ(p) ⎞ ⎟ ⎠ f(q) dμ(q) =  (T_μ,∗ g)f dμ.

ItiseasytocheckthatK∗satisfiesthegrowthconditionin(2.5).Moreover,K∗satisfies the Hölder continuity requirement in (2.5) with exponent β/2; this is a corollary of Lemma2.1.

2.2.1. Two examples

Inthisshortsection, wegivetwo examplesof concrete3-dimensionalCZkernels. Example2.4(The 3-dimensionalH-Rieszkernel).Considerthekernel

K(z) = ∇Hz−2, z∈ H \ {0}. (2.10)

Notethat· −2agrees(uptoamultiplicativeconstant)withthefundamentalsolution of the sub-Laplacian Δ_H, as proved by Folland [21], see also [4, Example 5.4.7]. We call K the 3-dimensional H-Riesz kernel; it gives rise to a SIO R, which we call the 3-dimensional H-Riesz transform. Studyingthe L2_{-boundedness of R on subsets of H}

isconnectedwith theremovabilityofthese setsforLipschitz harmonicfunctionsonH, seeTheorem5.1 fortheprecise statement.

Theneatformula(2.10) canbeexpandedtothefollowingratherunwieldyexpression: K(x, y, t) = −2x|(x, y)|2_{+ 8yt} (x, y, t)6 , −2y|(x, y)|2_{− 8xt} (x, y, t)6  =: (K1(x, y, t),K2(x, y, t)). (2.11) Fromtheformulaabove,oneseesthatK isnotantisymmetricintheusualsenseK(z−1)= −K(z); for instance, K1_{(0,1,1) = K}1_{(0,−1,−1). However, both componentsof K are}

horizontallyantisymmetric,asinthedefinitionbelow.

Definition2.5(Horizontalantisymmetry).AkernelK :H\{0}→ R iscalledhorizontally antisymmetric,if

K(x, y, t) =−K(−x, −y, t), (x, y, t)∈ H \ {0}. Itisclearfromtheformula(2.11) thatK ishorizontallyantisymmetric. Example2.6(The 3-dimensional quasiH-Rieszkernel).Consider

Ω(x, y, t) := x (x, y, t)4, y (x, y, t)4, t (x, y, t)5  , (x, y, t)= (0, 0, 0). (2.12) It is easy to see that Ω is a 3-dimensional CZ kernel which is antisymmetric in the sense thatΩ(p)=−Ω(p−1) for allp∈ H \ {0}.We willcall Ω the 3-dimensional quasi H-Riesz kernel; it defines the 3-dimensional quasi H-Riesz transform Q. The kernel Ω, whichresembles informthe Euclidean Riesz kernels,was introduced in[11]. Itwas proved there thatif μ isa 3-ADR measure and Q is bounded inL2(μ) thenspt μ can be approximatedatμ almosteverypointandatarbitrarysmallscalesbyhomogeneous subgroups.It isunknowniftheH-Riesztransformhasthesameproperty.

2.2.2. Cancellationconditions

Fix a 3-dimensional CZ kernel K :H \ {0} → R. Without additional assumptions, the SIO T associatedwith K isgenerally notbounded on L2_{(μ), even when μ isnice,}

such as the 3-dimensional Hausdorffmeasure ona vertical plane W (see Section2.3.1 for a definitionof these planes), which is a constantmultiple of Lebesgue measure on W . So,forpositiveresults,oneneedstoimpose cancellationconditions.Thehorizontal antisymmetry or antisymmetry would be such conditions, but neither of them holds both for the H-Riesz kernel and the quasi H-Riesz kernel simultaneously. Here is a moregeneralcancellationcondition,whichencompassesantisymmetricandhorizontally antisymmetric kernels:

Definition2.7(AB).AkernelK :H \ {0}→ R satisfiestheannularboundedness condi-tion(ABforshort)ifthefollowingholds. Forevery· -radial C∞functionψ :H → R satisfyingχB(0,1/2)≤ ψ ≤ χB(0,2),thereexists aconstantAψ≥ 1 suchthat

    W [ψR(w)− ψr(w)]K(w) dL2(w)   ≤ Aψ (2.13)

forall0< r < R <∞,and forallvertical planesW .Above, ψr(z) := (ψ◦ δ1/r)(z),

where δr isthe(· -homogeneous)dilatationδr(x,y,t)= (rx,ry,r2t) for (x,y,t)∈ H.

It turns outthat theAB condition for3-dimensional CZ kernels K isequivalent to thefollowingcondition:

Definition2.8 (UBVP).Given akernel K :H \ {0}→ R withK(z) z−3,we say thatitisuniformlyL2 _{boundedonverticalplanes (UBVPinshort),iftheSIOassociated}

toK isboundedonL2_(H3_|

W) foreveryverticalplaneW (inthesenseofDefinition2.2),

withconstantsindependentof W . The measure H3_|

W is 3-ADR, so it makes sense to discuss boundedness of T on

L2(H3|_W).Thefollowing lemmaisananalogof[34,Proposition2,p.291].

Lemma 2.9. Assumethat akernel K :H \ {0}→ R with K(z) z−3 satisfies the UBVPcondition.Then K alsosatisfies theABcondition.

Proof. Fixavertical planeW ⊂ H.Thegroupoperation “·”restrictedto W coincides withusual(Euclidean)additionintheplaneW :ifv,w∈ W ,thenv−1· w = w − v.Also, H3_|

W = c· L2 forsomepositiveconstantc.Hence,forw∈ W ,

TH3_|W,f (w) = c



W

K(w− v)B(w− v)f(v) dL2(v) = c(KB)∗ f(w), f ∈ C0∞(W ),

where B is the indicator function of W \ B(0,), the notation “∗” means Euclidean

convolution,andC₀∞(W ) stands forsmooth andcompactlysupportedfunctionsonW . SinceT_H3_|W,is L2 boundedonW ∼=R2,itfollows thattheFouriertransform ofKB

isabounded functiononW , independentlyof:

|KB(ξ)| ≤ A, ξ∈ W ,  > 0.

For theproof see e.g.[25, 2.5.10]. Now, fix a functionψ :H → R as in Definition2.7. Fixalso 0< r < R < ∞, and avertical planeW . Note that ψR_{− ψ}r _{vanishes inthe}

ballB(0,r/2).Hence,if0< < r/2,wehaveψR(w)− ψr(w)= [ψR(w)− ψr(w)]B(w)

forw∈ W ,andhence   [ψR(w)− ψr(w)]K(w) dL2(w) =  [ψR(w)− ψr(w)]K(w)B(w) dL2(w)    A  ψR_(ξ)− _ψr_{(ξ) dL}2_(ξ),

usingPlancherelbeforepassingtothesecond line.Moreover,  | ψR_(ξ)| dL2_{(ξ) =}  | ψ(δR(ξ))|R3dL2(ξ) =  | ψ(ξ)| dL2_{(ξ) 1,}

andthesameholds with“r”inplaceof“R”.Thiscompletestheproof. 2

Now,recallthemainresult,Theorem1.1.Withtheterminologyabove,itstatesthat ifa3-dimensionalCZkernelsatisfiestheUBVP,thentheassociatedSIOisboundedon certain L2 _{spaces, whichwe will define momentarily (seeSection 2.3). Thestrategy of}

proof istoinfer,fromthelemmaabove,thatthekernel satisfiestheABcondition,and proceedfromthere.Inparticular,itremainstoprovethefollowingversionTheorem1.1: Theorem 2.10.Letα > 0,and assumethat φ∈ C1,α_{(W ) has compactsupport. Assume}

that a 3-dimensionalCZ kernel K :H \ {0}→ R satisfies theAB condition.Then, the associatedSIOisboundedonL2_{(μ) forany3-Ahlfors-Davidregularmeasureμ supported}

on theintrinsicgraph ofφ.

As simplecorollaries, theH-Riesz transformR and thequasiH-Riesz transform Q, recallSection2.2.1,areL2_{boundedontheintrinsicgraphsmentionedinTheorem2.10.}

Since the associated kernels K and Ω are 3-dimensional CZ kernels, it suffices to ver-ify that they satisfy the AB condition. But this is a consequence of either horizontal antisymmetry (in the case of K) or antisymmetry (in the case of Ω). In fact, the key cancellationcondition(2.13) evenholdsinthestrongerform



[ψR(w)− ψr(w)]K(w) dL2(w) = 0

forallfunctionsψ asinDefinition2.7,forall0< r < R <∞,andallverticalplanesW . 2.3. Intrinsic graphsandtheboundedness ofSIOs

For which 3-ADR measures μ are the SIOs associated to 3-dimensional CZ kernels satisfying the UBVP condition bounded on L2_{(μ)? The following seems likea natural}

conjecture:

Conjecture 2.11.Let W ⊂ H be a vertical subgroup with complementary subgroup V , and let φ :W → V be an intrinsic Lipschitz function (see Definition 2.12). If T is a convolutiontypeSIOwitha3-dimensionalCZkernelwhichsatisfiestheUBVPcondition, then itisbounded onL2_{(μ) forall3-ADR measures μ supportedon theintrinsic graph}

Γ(φ). In particular, this is true for μ = H3_|

Γ(φ) (since H3|Γ(φ) is 3-ADR by Theorem

3.9in [22]).

Recall thatthemain theorem ofthe paper, Theorem1.1, statesthat theconjecture holdsforφ :W → V ,whicharecompactlysupportedandintrinsicallyC1,α_{(W )-smooth}

forsomeα∈ (0,1],seeDefinition2.16. 2.3.1. IntrinsicLipschitz graphs

In our terminology, the vertical subgroups in H are all the nontrivialhomogeneous normal subgroups of H, except for the centre of the group. Recall that homogeneous

subgroups are subgroups of H which are preserved under dilations of the Heisenberg group,see[33].Withthechoiceofcoordinatesasin(2.1),theverticalsubgroupscoincide thereforewiththe2-dimensionalsubspacesofR3_{thatcontainthet-axis.Toeveryvertical}

subgroup W we associate acomplementary horizontal subgroup V . In our coordinates thisissimplythe1-dimensionalsubspaceinR3whichisperpendiculartoW .Everypoint p∈ H can be written as p = pW · pV with a uniquelydetermined vertical component

pW ∈ W andhorizontalcomponentpV ∈ V .ThisgivesrisetotheHeisenbergprojections

πW :H → W , πW(p) = pW

and

π_V :H → V , π_V(p) = p_V. Definition2.12.Anintrinsic graph isasetoftheform

Γ(φ) ={w · φ(w) : w ∈ W },

where W ⊂ H isa vertical subgroupwith complementary horizontalsubgroup V , and φ :W → V is any function. We often use the notation Φ for the graph map Φ(w) = w· φ(w).To define intrinsic Lipschitz graphs, fix aparameter γ > 0,and consider the set(cone)

Cγ={z ∈ H : πW(z) ≤ γπV(z)}.

We say that φ is an intrinsic L-Lipschitz function, and Γ(φ) an intrinsic L-Lipschitz graph,if

(z· Cγ)∩ Γ(φ) = {z}, for z ∈ Γ(φ) and 0 < γ < L1.

The function φ is said to be intrinsic Lipschitz if it is intrinsic L-Lipschitz for some constantL≥ 0.

Remark2.13.Everyvertical subgroupW canbeparametrisedas W = {(−w1sin θ, w1cos θ, w2) : (w1, w2)∈ R2}

with an angle θ ∈ [0,π) uniquely determined by W . The complementary horizontal subgroupisthengivenby

V = {(v cos θ, v sin θ, 0) : v ∈ R}.

We often denote points on W in coordinates by “(w1,w2)”, and points on V byreal

numbers “v”. Then, expressions such as (w1,w2)· (w1,w2) and (w1,w2)· v should be

interpretedas elementsinH,namelytheproductsofthecorrespondingelementsonW andV .

Intrinsic Lipschitz graphswere introduced byFranchi, Serapioni and Serra Cassano in [24], motivated by the study of locally finite perimeter sets and rectifiability inthe Heisenberg group [23]. While intrinsic Lipschitz functions continue to be studied as a classofmappingswhichareinterestingintheirownright,theyhavealsorecentlyfound a prominent application in [29]. Various properties of intrinsic Lipschitz functions are discussed indetailin[33]. Forinstance,itisknownthatanintrinsicLipschitz function has a well-defined intrinsic gradient ∇φφ ∈ L∞(H3|_W), which we will use to perform integration onintrinsicLipschitzgraphs.

2.3.2. Intrinsicdifferentiability

To define the intrinsic gradient, we recall that the notion of intrinsic graph is left invariant. Indeed, given a function φ : W → V with intrinsic graph Γ(φ), for every p ∈ H, the set p· Γ(φ) is again the intrinsic graph of a function W → V , which we denotebyφp_{,sothatp·Γ(φ)= Γ(φ}p_{).ForinstanceifW isthe(y,t)-plane,V thex-axis,}

andp0= (x0,y0,t0),thenwe cancomputeexplicitly

φp0(y, t) = φ(y− y0, t− t0+1₂x0y0− yx0) + x0. (2.14)

Wealso recall thatinour context anintrinsic linearmap isafunction G: W → V whoseintrinsicgraphisaverticalsubgroup.

Definition2.14.A functionψ :W → V with ψ(0)= 0 isintrinsically differentiableat0 ifthereexists anintrinsiclinearmap G:W → V suchthat

(Gw)−1_{· ψ(w) = o(w), as w → 0. (2.15)}

ThemapG iscalled theintrinsicdifferential ofψ at0 anddenotedbyG= dψ0. Moregenerally,afunctionφ:W → V isintrinsicallydifferentiableatapointw0∈ W

ifψ := φ(p−10 )_{isintrinsicdifferentiableat}0 for_{p0:= w0}·φ(w_{0).Theintrinsicdifferential}

of φ atw0 isgivenby

dφw0 := dψ0.

Recall that V can be identified with R through our choice of coordinates, see Re-mark2.13.UnderthisidentificationtherestrictionoftheKorányi distanceto V agrees withtheEuclidean distance|· | so that(2.15) reads

|ψ(w) − Gw| = o(w), as w → 0.

WiththeparametrisationfromRemark2.13,everyintrinsiclinearmapG:W → V has theformG(w1,w2)= cw1foraconstantc∈ R.

Definition2.15.Assumethatφ:W → V isintrinsicallydifferentiableatapointw0∈ W .

Thenitsintrinsicgradientatw0 istheuniquenumber∇φφ(w0) suchthat

dφw0(w1, w2) =∇φφ(w0)w1, for all (w1, w2)∈ W .

Theintrinsicgradient∇φ_φ(w

0) issimplyanumberdeterminedbythe“angle”between

W andtheverticalplanedφw0(W ).

2.3.3. C1,α_{-intrinsic Lipschitzfunctions andgraphs}

The goal of the paper is to prove that certain SIOs are bounded in L2 _on

intrin-sic graphs Γ(φ), where φ is a compactly supported function satisfying a (Heisenberg analogueof) C1,α_{-regularity. The most obvious definition of C}1,α _{would be to require}

the intrinsicgradient ∇φφ to be locally α-Hölder function inthe metricspace (W ,d), butthiscondition isnotleft-invariant:theparametrisationofthe left-translatedgraph p−1· Γ(φ),for p∈ Γ(φ),wouldnotnecessarily belocally α-Hölder continuous withthe sameexponentα,seeExample 4.5.So,instead,wedefine an“intrinsic”notionofC1,α_,

whichis (a) left-invariant inthesense above,and (b)is well-suitedfor the application wehaveinmind,and(c)isofteneasytoverify,seeRemark2.21below.

Definition 2.16.Wesay that afunction φ: W → V isan intrinsic C1_{(W ) function if}

∇φ_{φ exists at everypoint w∈ W , andis continuous. We furtherdefine thesubclasses}

C1,α_{(W ),α∈ (0,1],asfollows:φ∈ C}1,α_{(W ),ifφ∈ C}1_{(W ),andthereexistsaconstant}

H ≥ 1 suchthat

|∇φ(p−10 )_φ(p−1₀ )_{(w)− ∇}φ(p−10 )_φ(p−1₀ )_{(0)| ≤ Hw}α_{, (2.16)}

forallp0∈ Γ(φ),andallw∈ W .Fornotationalconvenience,wealsodefineC1,0(W ):=

C1_{(W ). IntrinsicgraphsofC}1 _(orC1,α_{)functionswillbe called intrinsicC}1 _{(or C}1,α₎

graphs.

Severalremarksarenowinorder.

Remark2.17.(a)Itiswell-known,seeforinstanceProposition4.4in[14] orLemma 4.6 in[8],thatifφ∈ C1_{(W ) isintrinsicLipschitz, then∇}φ_{φ∈ L}∞_{(W ).}

(b)Conversely,ifφ∈ C1_{(W ) with∇}φ_{φ∈ L}∞_{(W ),thenφ isintrinsicLipschitz.This}

iswell-knownandfollowsfromexistingresults,butitwasdifficulttofindareferenceto thisparticularstatement;henceweincludetheargument inLemma 2.22below. Remark2.18.Note thatifφ∈ C1_{(W ) has compactsupport,then ∇}φ_{φ∈ L}∞_{(W ), and}

henceφ isintrinsicLipschitzby(b)above.

Remark2.19.IfW isthe(y,t)-plane,thecondition(2.16) forp0= (y0,t0)· φ(y0,t0) can

bewrittenincoordinatesasfollows:

|∇φ_φ_{y + y}

0, t + t0+ φ(y0, t0)y



− ∇φ_φ(y

0, t0)| ≤ H(y, t)α. (2.17)

Tosee this,applytherepresentation(2.14) withp0 replacedbyp−10 .

Remark2.20.ItisknownbyTheorem4.95in[33] thattheintrinsicgraphofanintrinsic C1_{(W ) functionis an H-regularsurface;in particular,φ satisfies anarea formula, see}

Section2.3.4formoredetails.

Thedefinitionsof(intrinsic)C1_{(W ) andC}1,α_{(W ) arequitedifferentfromtheir}

stan-dard Euclidean counterparts, which we denote by C1_(R2_{) and C}1,α_(R2_{) (a function}

belongs to C1,α_(R2_{) if its partial derivatives exist and are α-Hölder continuous with}

respect to theEuclidean metric).However,at leastfor compactlysupportedfunctions, sufficientregularityintheEuclideansensealsoimpliesregularityintheintrinsic Heisen-bergsense,asthefollowingremark shows.

Remark 2.21.Assumethat W is the (y,t)-plane, and identifyW with R2. Then, any compactly supportedC1,α(R2)-function isin theclass C1,α(W ). Indeed,if φ∈ C1(R2) then∇φφ hasthefollowing expression:

∇φ_{φ = φ}

y+ φφt, (2.18)

see[8,(4.4)].Sinceφ,φtarebounded,φ isC1(R2) andφy,φtareEuclideanα-Hölder,we

inferthat∇φφ isEuclideanα-Hölder continuous.Sincealsoφy isbounded,weobtain

|∇φ_φ_{y + y}

0, t + t0+ φ(y0, t0)y



− ∇φ_φ(y

0, t0)|  min{1, |(y, t + φ(y0, t0)y)|α}

 min{1, |(y, t)|α_{}  (y, t)}α_,

whichby(2.17) verifiesthatφ∈ C1,α_{(W ).}

Lemma 2.22.Assume that φ∈ C1_{(W ) with ∇}φ_{φ∈ L}∞_{(W ).Then, φ isintrinsic}

Lips-chitz.

Proof. For simplicity, we assume thatW is the(y,t)-plane. WriteL :=∇φ_φ L∞(W ).

ByProposition4.56(iii)in[33],itsufficesto verifythat |φ(p−1)_{(y, t)| = |φ}(p−1)_{(y, t)− φ}(p−1)_{(0, 0)| }

L(y, t), (y, t)∈ W , p ∈ Γ(φ).

Writep= w· φ(w).Then, weestimateasfollows:

|φ(p−1)_{(y, t)| ≤ |φ}(p−1)_{(y, t)− ∇}φ_{φ(w)y| + |∇}φ_φ(w)y|

L(y, t) + L(y, t),

as claimed. The estimate leading to the last line follows from Proposition 2.23 (with α = 0)below. 2

Proposition 2.23.Fix α ∈ [0,1], and assume that W is the (y,t)-plane. Assume that φ∈ C1,α_{(W ) withL:=∇}φ_φ

L∞(W )<∞.Then, forp= w· φ(w)∈ Γ(φ),

|φ(p−1)_{(y, t)− ∇}φ_{φ(w)y|  (y, t)}1+α_{, (y, t)∈ W ,}

wheretheimplicit constants onlydepend on L,and, if α > 0, also ontheHölder conti-nuity constant“H” in thedefinition ofC1,α_{(W ).}

WepostponetheprooftoSection4.

Remark2.24.If α > 0, Proposition2.23 aboveshows that functionsin C1,α_{(W ) with}

∇φ_{φ∈ L}∞_{(W ) areuniformly intrinsicallydifferentiable, see[2, Definition3.16].}

ThenextlemmaverifiesthatbeinganintrinsicC1,α_{-graphisaleft-invariantconcept.}

Lemma2.25.Letα > 0.Letφ∈ C1,α_{(W ) withconstant “H” asinDefinition 2.16,and}

writeΓ:= Γ(φ).Fixq∈ H,andconsiderthefunctionφ(q),whichparametrisesthegraph q· Γ. Then, φ(q)_{∈ C}1,α_{(W ) withthesameconstant “H”.}

Proof. Letψ be themapφ(q) which parametrisesthetranslatedgraphq· Γ. Sinceφ is byassumptioneverywhereintrinsicallydifferentiable,andsinceintrinsicdifferentiability isaleft-invariantnotion,ψ isintrinsicallydifferentiableeverywhere.Itremainstoverify foreveryp0∈ Γ(ψ) andforallw∈ W that

|∇ψ(p−10 )

ψ(p−10 )_(w)− ∇ψ(p−10 )_ψ(p−10 )₍0)| ≤ Hwα_{. (2.19)}

Bydefinitionψ(p−10 )_{= (φ}(q)₎(p−10 ) _{parametrisesthegraphp}−1

0 · q · Γ andhence

ψ(p−10 )_{= φ}(p−10 ·q)_{= φ}([q−1·p0]−1)_.

Thus,denotingp:= q−1· p0,theexpressionwewishto estimate,readsas follows:

|∇ψ(p−10 )

ψ(p−10 )_(w)− ∇ψ(p−10 )_ψ(p−10 )₍0)| = |∇φ(p−1)_φ(p−1)_(w)− ∇φ(p−1)_φ(p−1)₍0)|.

Since p= q−1· p0 is apoint in q−1· Γ(ψ) = q−1· q · Γ = Γ, the estimate (2.19) then

followsfromtheassumptionφ∈ C1,α_{(W ),moreprecisely(2.16) appliedwithp instead}

ofp0. 2

2.3.4. Areaformula

Wewillneed anareaformula forfunctionsφ∈ C1_{(W ).Since theintrinsicgraphsof}

suchfunctionsareH-regularsubmanifoldsbyRemark2.20, suchaformulaisavailable, duetoAmbrosio,SerraCassanoandVittone[1].Specialisedtooursituation,theformula readsas follows:thereexists ahomogeneousleft-invariantmetricd1onH suchthat

S3 d1(Φ(Ω)) =  Ω  1 + (∇φ_φ)2_dL2 _(2.20)

for all φ ∈ C1_{(W ) with graph map Φ, and all open sets Ω ⊂ W . Here S}3

d1 is the

3-dimensionalsphericalmeasuredefinedviathedistanced1.Thisisessentiallythearea

formulawewereafter,butwewill stillrecordthefollowinggeneralisation:

Proposition 2.26.Thereexistsa left-invarianthomogeneous distanced1 on H such that

theassociatedspherical HausdorffmeasureS3_=S3

d1 satisfies  Γ h dS3₌  W (h◦ Φ)  1 + (∇φ_φ)2dL2 _(2.21)

forevery φ∈ C1_{(W ) withgraph Γ, andforeveryh∈ L}1_(S3_| Γ).

Proof. Thisis standard: forany bounded open set Ω⊂ W , formula (2.20) implies the claim for the functionh = χΦ(Ω). Theformula for arbitrary L1-functions h followsby

approximation. 2

3. Proofofthemain theorem

In this section, we prove the main result, Theorem 1.1, or rather its variant, The-orem 2.10; recall that this is sufficient by the discussion preceding the statement of Theorem 2.10.Wefixthefollowing data:

• a3-dimensionalCZkernelK satisfyingtheABcondition,

• averticalsubgroup W withcomplementaryhorizontalsubgroupV (wewillassume withoutloss ofgeneralitythatW isthe(y,t)-plane andV isthex-axis),

• afunctionφ∈ C1,α_{(W ),α > 0,with compactsupport(recallDefinition2.16),and}

• a3-ADRmeasure μ onΓ:= Γ(φ).

ThetaskistoshowthatthesingularintegraloperatorT associatedtoK isboundedon L2_(μ):

Tμ,fL2(μ)≤ AfL2(μ), f ∈ L2(μ),  > 0, (3.1)

whereA≥ 1 isaconstantdependingonthedataabove,butnoton.Thefirstreduction isthefollowingeasylemma:

Lemma3.1. Assumethat(3.1) holdsforsome3-ADRmeasureμ onΓ.Then(3.1) holds (with possiblyadifferentconstant)forany 3-ADR measureμ with˜ spt ˜μ⊂ spt μ.

Proof. Sinceμ is3-ADR,weclearlyhaveμ= ϕμdH3|Γ forsomeϕμ ∈ L∞(H3|Γ).Recall

thatH3_|

Γ is 3-ADR by [22, Theorem 3.9] since Γ is the intrinsic graphof an intrinsic

Lipschitz function according to Remark 2.18. Therefore, (Γ,H3_{) is a doubling metric}

measurespace, whereLebesgue’sdifferentiationtheoremholds.Thuswe alsohavethat 1 lim sup r→0 μ(B(p, r)) H3_{(Γ∩ B(p, r))} = lim sup r→0 1 H3_{(Γ∩ B(p, r))}  B(p,r)∩Γ ϕμdH3= ϕμ(p)

forH3_{almost everyp∈ spt μ,and inparticularforμ almost everyp∈ spt μ.So,inthe}

metricmeasure space(Γ,H3_{), wehaveϕ}

μ∼ χspt μ. Thesameholdsfor μ,˜ bythesame

argument.Sincespt ˜μ⊂ spt μ,itfollowsthatwemaywriteμ = g dμ for˜ someg∈ L∞(μ) withg∼ χspt ˜μ.Withthisnotation,wehave

Tμ,˜ f (p) =  q−1_·p> f (q)K(q−1· p) d˜μ(q) =  q−1_·p> f (q)g(q)K(q−1· p) dμ(q) = Tμ,(f g)(p). Finally, Tμ,˜ f (p)L2(˜μ) Tμ,(f g)L2_(μ)≤ Afg_L2_(μ)= Af_L2(˜μ), asclaimed. 2

The point of the lemma is thatif we manage to prove (3.1) for any single 3-ADR measureμ with spt μ= Γ,thenthesamewill followforall3-ADR measuressupported onΓ.Inparticular,itsufficesto prove(3.1) forthemeasure

μ :=S_d13 |Γ, (3.2)

whichsatisfiestheareaformula, Proposition2.26.

Forthismeasureμ,weprove(3.1) byverifyingtheconditionsofasuitableT 1 theorem. Tostatetheseconditions,weuseasystemofdyadiccubesonΓ.

3.1. Christ cubesand theT 1 theorem

Thefollowing constructionis dueto Christ[13]. Forj ∈ Z,there exists afamilyΔ_j ofdisjointsubsetsofΓ withthefollowingproperties:

(C0) Γ⊂_Q∈Δ jQ,

(C1) Ifj ≤ k,Q∈ Δj andQ∈ Δk,then eitherQ∩ Q=∅,or Q⊂ Q.

(C2) IfQ∈ Δj,thendiam Q≤ 2j =: (Q).

(C3) Every cube Q ∈ Δj contains a ball B(zQ,c2j)∩ Γ for some zQ ∈ Q, and some

constantc> 0.

(C4) Every cube Q ∈ Δj has thin boundary: there is a constant D ≥ 1 such that

μ(∂ρQ)≤ Dρ

1

D_μ(Q),where

∂ρQ :={q ∈ Q : dist(q, Γ \ Q) ≤ ρ · (Q)}, ρ > 0.

Thesets inΔ:=∪Δj arecalled Christ cubes (sometimes alsoDavid cubes inthe

liter-ature), or justdyadiccubes, ofΓ.It followsfrom(C2), (C3), andthe3-regularity ofμ thatμ(Q)∼ (Q)3 _{forQ∈ Δ}

j.

To provethe L2 _{boundedness of aCZ operatorT on L}2_{(μ),it suffices to verifythe}

following conditionsforafixedsystemΔ ofdyadiccubesonΓ:

Tμ,χR2L2_(μ|R)≤ Aμ(R) and Tμ,∗ χR2L2_(μ|R)≤ Aμ(R) (3.3)

for all R ∈ Δ, where T_μ,∗ is the formal adjointof Tμ,, and A ≥ 1 is aconstant

inde-pendentof  andR. Theseconditionssuffice for(3.1) by theT 1 theoremof Davidand Journé, applied in the homogeneous metric measure space (Γ,d,μ), see [36, Theorem 3.21]. The statement in Tolsa’s book is only formulated in Euclidean spaces, but the proof works the same way inhomogeneous metric measure spaces; the details can be foundinthehonorsthesisofSurathFernando[20].

3.2. VerifyingtheT 1 testingcondition

Inthis section,we usetheT 1 theoremto proveTheorem2.10(henceTheorem 1.1). Thenotationα,K,φ,Γ,W referstothedatafixedattheheadofSection3,andμ isthe measure in(3.2). We start by remarking that it is sufficient to verify the first testing condition,namely

Tμ,χR2L2_(μ|R)≤ Aμ(R), for all R ∈ Δ. (3.4)

This followsfromthesimpleobservationthatT∗ hasthesameformas T ,so ourproof belowfor T would equallywell work forT∗. Letus be abit more precise. Remark2.3 shows that the kernel K∗ associated with the formal adjoint T∗ is also a CZ kernel. Moreover, since (i) the functions ψ appearing in Definition 2.7 are radial, (ii) w = w−1_{,and(iii)themeasureL}2_{isinvariantunderthetransformationw→ w}−1 _{onW ,}

itfollowsthatK∗ satisfiestheABconditionwith thesameconstantasK.

Thefirststepintheproofof(3.4) isaLittlewood-Paleydecompositionoftheoperator T ;thedetailsintheHeisenberggroupappearedin[9],andwecopythemnearlyverbatim. Fixasmoothevenfunctionψ :R→ R withχB(0,1/2)≤ ψ ≤ χB(0,2),andthendefinethe

(H-)radialfunctionsψj:H → R by

ψj(p) := ψ(2jp), j∈ Z.

Next,writeηj := ψj− ψj+1 andK(j):= ηjK,so that

spt K(j)⊂ B(0, 21−j)\ B(0, 2−2−j). (3.5)

Wenowconsidertheoperators T(j)f (p) =  K(j)(q−1· p)f(q) dμ(q) and SN :=  j≤N T(j),

Remark3.2. ItiseasytocheckthatthekernelK(j)satisfiesthesamegrowthandHölder

continuity estimates, namely (2.5), as K. In particular, by (3.5) K(j) ∈ L∞(H) with

K(j)L∞(H) 23j.

The next lemma demonstrates that Tμ, and SN are very close to each other, for

∈ [2−N,2−N+1):

Lemma3.3. Fix N∈ Z and ∈ [2−N,2−N+1). Then

|SNf (p)− Tμ,f (p)|  Mμf (p), f ∈ L1loc(μ),

whereMμ isthecentredHardy-Littlewood maximalfunctionassociated withμ.

Proof. Wefirstobservethat  j≤N K(j)(p) = K(p)  j≤N ηj(p) = K(p)· (1 − ψN +1(p)), p∈ H \ {0}. Hence,for∈ [2−N,2−N+1), |SNf (p)− Tμ,f (p)| =   K(q−1· p)[(1 − ψN +1)− χH\B(0,)](q−1· p)f(q) dμ(q)   ≤  B(p,2−N+1)\B(p,2−N−2) |K(q−1_{· p)||f(q)| dμ(q)}  1 μ(B(p, 2−N+1))  B(p,2−N+1) |f(q)| dμ(q)  Mμf (p),

usingthegrowthcondition(2.5). 2

Bythelemmaabove,andtheL2_{-boundednessofM}

μ,wehave

Tμ,χRL2_(μ|R) MμχRL2_(μ)+S_Nχ_R_L2_(μ|R) μ(R)1/2+SNχRL2_(μ|R).

So,itremainstoprovethefollowingproposition:

Proposition 3.4.Forany R∈ Δ andN ∈ Z,wehave SNχR2_L2_(μ|R) μ(R).

Proof. Wefix R∈ Δ andN ∈ N fortherestoftheproof. Westartwith theestimate SNχRL2_(μ|R)≤  2−N_≤2−j_{≤4 (R)} T(j)χR L2(μ|R) +  2−j_{>4 (R)} T(j)χRL2_(μ|R) (3.6)

Wequicklydealwiththetermsinthesecondsum.Recallfrom(3.5) thatthesupportof q→ K(j)(q−1· p), p∈ R,

is containedin H \ B(p,2−2−j)⊂ H \ R, assuming2−j > 4(R).Hence T(j)χR(p) = 0

forallp∈ R and2−j > 4(R).So, 

2−j_{>4 (R)}

T(j)χRL2_(μ|R)= 0. (3.7)

Thus,itremainsto studytheterm

SχRL2(μ|R), where S :=



2−N≤2−j_{≤4 (R)} T(j).

WestressthattheoperatorS dependsbothonN andR,buttoavoidheavynotation,we refrainfromexplicitlymarkingthisdependence.Alltheimplicitmultiplicativeconstants whichappearinthefollowingestimateswill beuniforminN andR.

The strategy for bounding Sχ_R_L2_(μ|R) is straightforward: we fix a point p ∈ R,

and attempt to find an estimate for SχR(p). The quality of this estimate will depend

on thechoice ofp as follows: thecloser p is to the“boundary” ofR, inthe sense that dist(p,Γ\ R) issmall, theworsetheestimate.Motivatedbythisdiscussion,wedefine

∂ρR ={q ∈ R : (ρ/2) · (R) < dist(q, Γ \ R) ≤ ρ · (R)}, ρ > 0,

and recall thatμ(∂ρR) ρ

1

D_{μ(R) by} (C4)fromSection3.1 (the notationwe use here is a little different from Section 3.1, in that we impose a two-sided inequality in the definition of ∂ρR).Also note that ∂ρR = ∅ for ρ > 2. Write ρ(k) := 21−k/(R), and

decompose Sχ_R2 L2_(μ|R)as follows: SχR2L2_(μ|R)=  2−k_{≤ (R)}  ∂ρ(k)R |SχR(p)|2dμ(p) =:  2−k_{≤ (R)} Ik.

The task is now to estimate theterms Ik separately. Note that whenever p ∈ ∂ρ(k)R,

then

T(j)χR(p) =  R K(j)(q−1· p) dμ(q) =  Γ K(j)(q−1· p) dμ(q), j > k, (3.8)

since the support of q → K(j)(q−1 · p) lies in B(p,21−j) ⊂ B(p,2−k) and moreover,

dist(p,Γ\ R)≥ 2−k for p∈ ∂ρ(k)R,thereforeB(p,2−k)⊂ R for suchp.Wewould now

liketosplit theoperatorS intotwo parts–depending onk: asumoftermswhere the estimate(3.8) is valid(corresponding toindices j with 2−j ≤ 2−k),and asumwith all the remaining terms. However, if 2−k > 1, we wish to further split the first sum into two parts, which we will discuss separately. Thus let us momentarily fix k ∈ Z with 2−k≤ (R) and considerthefollowingk-dependentsplitting:

S = SI+ SII + SIII=  j∈E1(k) T(j)+  j∈E2(k) T(j)+  j∈E3(k) T(j), where E1(k) :={j ∈ Z : 2−N ≤ 2−j < min{1, 2−k}} E2(k) :={j ∈ Z : min{1, 2−k} ≤ 2−j < 2−k} E3(k) :={j ∈ Z : 2−k≤ 2−j ≤ 4(R)}.

If2−k≤ 1,thenE2(k)=∅ andwesimplyhaveS = SI+ SIII.Hence,

 2−k≤ (R) Ik   2−k≤ (R)  ∂ρ(k)R |SIχR(p)|2dμ(p) (3.9) +  1<2−k_{≤ (R)}  ∂ρ(k)R |SIIχR(p)|2dμ(p) (3.10) +  2−k_{≤ (R)}  ∂ρ(k)R |SIIIχR(p)|2dμ(p) (3.11)

Onallthreelines,weneedtoget μ(R) ontherighthandside;westartwithline(3.11). ThepointwiseestimatewecanobtainforSIIIχR(p),forp∈ ∂ρ(k)R,isfairlylousy:itis

basedonthetrivialestimate

|T(j)f (p)|  fL∞_(μ), j∈ Z, p ∈ H, f ∈ L∞(μ). (3.12) Thisfollowsbyobservingthatthesupportof

q→ K(j)(q−1· p) (3.13)

iscontainedintheannulusB(p,21−j_{)\ B(p,2}−2−j_{),so |K}

(j)(q−1· p)| 23j,andfinally

|T(j)f (p)|  fL∞_(μ) 

B(p,21−j)

23jdμ fL∞_(μ).

Inparticular,(3.12) impliesthat |SIIIχR(p)|  C + log

(R)

2−k ∼ C + log 1

ρ(k), p∈ ∂ρ(k)R. Recallingagainthatμ(∂ρR) ρ

1 Dμ(R), weobtain  ∂ρ(k)R |SIIIχR(p)|2dμ(p) ρ(k) 1 D C + log 1 ρ(k) 2 μ(R).

Thelasttermsaresummableto  μ(R) overtherange2−k≤ (R) (whichisequivalent to ρ(k)≤ 2), sowearedonewithestimating line(3.11).

InestimatingSI andSII,thefollowingobservationiscrucial.Since(3.8) holdsforall

2−j < 2−k,itsanaloguealsoholds forSI andSII:

SIχR(p) =  Γ KI(q−1· p) dμ(q) and SIIχR(p) =  Γ KII(q−1· p) dμ(q) forp∈ ∂ρ(k)R,where KI :=  j∈E1(k) K(j) and KII :=  j∈E2(k) K(j).

Wewillnow dealwithline(3.10).Fixk with2−k> 1.Weclaimthatifp∈ ∂_ρ(k)R\ B(0,C), where C = C(φ) ≥ 1 is a suitable constant, then |SIIχR(p)| is bounded by

another constant,whichonlydependson theannularboundednesscondition. Thiswill givetheestimate

 ∂ρ(k)R |SIIχR(p)|2dμ(p)  B(0,C) |SIIχR(p)|2dμ(p) + μ(∂ρ(k)R), (3.14)

whichwillbe goodenough.

Now, assumethat∂ρ(k)R\ B(0,C)= ∅,andfixp= w0· φ(w0)∈ ∂ρ(k)R\ B(0,C).If

C waschosenlargeenough,thecompactsupportofφ impliesthat p = w0∈ W .

By(3.8) andtheareaformula,Proposition2.26,

SIIχR(p) =  Γ KII(q−1· p) dμ(q) =  W KII(ΦΓ(y, t)−1· w0)  1 +∇φ_{φ(y, t)}2_{dy dt.}

Write Φ_W(w) = w for the graph map parametrizing W itself. Then, by the annular boundednessassumption,and theareaformulaagain,itfollows that

    W KII(ΦW(y, t)−1· w0) dy dt   =     W KII(w−1· w0) dS3(w)    =     W KII(w) dS3(w)    ≤A. (3.15)

Tojustifythelastinequality,wewrite     W KII(w) dS3(w)   =     W  j∈E2(k) K(j)(w) dS3(w)    =     W  k<j≤0 ηj(w)K(w) dS3(w)    =     W (ψ(2−(k+1)w) − ψ(w))K(w) dS3(w)   .

Observethatthefunctionψ :H → R, definedbyψ(w)= ψ(w), satisfies the condi-tionsfromDefinition2.7and ψ r(w)= ψ(δr−1(w))= ψ(w/r) for w∈ W and r > 0.

Thecareful reader mayhavenoticed thatDefinition2.7 has been formulated interms ofan integralwithrespect to the2-dimensional Lebesguemeasure L2, ratherthan S3. However,restrictedto W ,HeisenberggroupmultiplicationbehaveslikeadditioninR2_,

see (2.1), and so L2 yields a uniformly distributed measure on (W ,d). Since also S3 restrictedto W isauniformlydistributed measure,thetwo measuresL2 _andS3 _agree

upto amultiplicativeconstant, see Theorem 3.4 in[27]. Moreover, this constant does notdepend onthe choice of W since rotationsaround the vertical axis are isometries bothfortheEuclideanandtheKorányidistance.Inconclusion,(3.15) followsby(2.13). Consequently, SIIχR(p) only differs in absolute value by ≤ A from the following

expression:  W  KII(ΦΓ(y, t)−1· w0)  1 +∇φ_{φ(y, t)}2− K II(ΦW(y, t)−1· w0)  dy dt. (3.16)

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Weimmediatelynotethattheintegrandvanishesidenticallyfor(y,t)∈ W \ spt φ,since ΦΓ(y,t)= ΦW(y,t) and ∇φφ(y,t) = 0 forsuch (y,t). Whatif (y,t)∈ spt φ? Note that

iftheintegranddoesnotvanish,then,bythedefinitionofKII, wehave

ΦΓ(y, t)−1· w0∈  1<2−j_≤2−k spt K(j) or ΦW(y, t)−1· w0∈  1<2−j_≤2−k spt K(j).

Recall from (3.5) that spt K(j) ⊂ B(0,21−j)\ B(0,2−2−j). Hence, ifΦΓ(y,t)−1· w0 ∈

spt K(j)forinstance,wehave

p = w0∈ B(ΦΓ(y, t), 21−j)\ B(ΦΓ(y, t), 2−2−j)⊂ B(ΦΓ(y, t), 21−j). (3.17)

Given that ΦΓ(y,t) always lies inthe fixed ball B(0,diam(ΦΓ(spt φ))), independent of

theparticularchoiceof(y,t)∈ spt φ,andp∈ H \ B(0,C),weinferthat(3.17) canonly occur for  1 indices j with 2−j > 1 (the particular indices naturally depend on the locationofourfixedpointp= w0).Finally,noticingthatKIIL∞  1 (seeRemark3.2

and note that E2(k) only contains negative indices j), we see that the expression in

(3.16) isboundedby L2(spt φ) 1.This provesthat|SIIχR(p)| 1,and establishes

(3.14).

Finally, we observe that for p∈ B(0,C), we have thetrivial estimate |S_IIχR(p)|

1+ log (R), using (3.12) (note that since 2−k > 1, also (R) ≥ 2−k > 1). Combined with(3.14),andμ(B(0,C)) 1,weget

 ∂ρ(k)R |SIIχR(p)|2dμ(p) (1 + log (R))2+ μ(∂ρ(k)R). Hence (3.10)  1≤2−k_{≤ (R)} [(1 + log (R))2+ μ(∂ρ(k)R)] (R)3∼ μ(R), as desired.

It remains to estimate the “mainterm” on line (3.9). The plan is simply to give a point-wise estimate for the integrand |S_IχR(p)|2, for p ∈ ∂ρ(k)R. Fix p∈ ∂ρ(k)R, and

recall,by(3.8) andthedefinitionofSI,that

SIχR(p) =  Γ KI(q−1· p) dS3(q) =  (p−1)·Γ KI(q−1) dS3(q). (3.18)

Since (p−1)· Γ isa graphwith thesame properties as Γ,we mayassume thatp =0. More precisely, inthis last partof the proof, we will only rely on theC1,α_-hypothesis

(andnotthecompactsupportofφ),whichisleft-translation invariantbyLemma 2.25.