TRACE THEOREM ON THE HEISENBERG GROUP ON HOMOGENEOUS HYPERSURFACES

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TRACE THEOREM ON THE HEISENBERG GROUP ON HOMOGENEOUS HYPERSURFACES HAJER BAHOURI, JEAN-YVES CHEMIN, AND CHAO-JIANG XU Abstract : We prove in this work the trace and trace lifting theorem for Sobolev spaces on the Heisenberg groups for homogenenous hypersurfaces. Resume : Dans ce travail, nous demontrons des theoremes de trace et de relevement pour les espaces de Sobolev sur le groupe de Heisenberg pour des hypersurfaces homogenes. Key words Trace and trace lifting, Heisenberg group, Hormander condition, Hardy's inequality. A.M.S. Classification 35 A, 35 H, 35 S. 1. Introduction In this work, we continue the study of the problem of restriction of functions that belongs to Sobolev spaces associated to left invariant vector fields for the Heisenberg group Hd initiated in [4]. As observed in [4], the case when d = 1 is not very different from the case when d ≥ 2, but the statement in this particular case are less pleasant. Thus, for the sake of simplicity, we shall assume from now on that d ≥ 2. Let us recall that the Heisenberg group is the space R2d+1 of the (non commutative) law of product w · w? = (x, y, s) · (s?, x?, y?) = ( x+ x?, y + y?, s+ s? + (y|x?)? (y?|x) ) .

  • trace lifting

  • z0 ?

  • l2 ≤

  • theoremes de trace et de relevement pour les espaces de sobolev

  • hardy inequality

  • sobolev spaces

  • ?2 dw

  • z0 ·


Published : Monday, June 18, 2012
Reading/s : 8
Origin : univ-rouen.fr
Number of pages: 12
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TRACETHEOREMONTHEHEISENBERGGROUPONHOMOGENEOUSHYPERSURFACESHAJERBAHOURI,JEAN-YVESCHEMIN,ANDCHAO-JIANGXUAbstract:WeproveinthisworkthetraceandtraceliftingtheoremforSobolevspacesontheHeisenberggroupsforhomogenenoushypersurfaces.Re´sume´:Danscetravail,nousde´montronsdesthe´ore`mesdetraceetderele`vementpourlesespacesdeSobolevsurlegroupedeHeisenbergpourdeshypersurfaceshomoge`nes.KeywordsTraceandtracelifting,Heisenberggroup,Ho¨rmandercondition,Hardy’sinequality.A.M.S.Classification35A,35H,35S.1.IntroductionInthiswork,wecontinuethestudyoftheproblemofrestrictionoffunctionsthatbelongstoSobolevspacesassociatedtoleftinvariantvectorfieldsfortheHeisenberggroupHdinitiatedin[4].Asobservedin[4],thecasewhend=1isnotverydifferentfromthecasewhend2,butthestatementinthisparticularcasearelesspleasant.Thus,forthesakeofsimplicity,weshallassumefromnowonthatd2.LetusrecallthattheHeisenberggroupisthespaceR2d+1ofthe(noncommutative)lawofproductww0=(x,y,s)(s0,x0,y0)=x+x0,y+y0,s+s0+(y|x0)(y0|x).Theleftinvariantvectorfieldsare1Xj=xj+yjs,Yj=yjxjs,j=1,∙∙∙,dandS=s=[Yj,Xj].2Inallthatfollows,weshalldenotebyZthisfamilyandstateZj=XjandZj+d=Yjforjin{1,∙∙∙,d}.Moreover,foranyC1functionf,weshallstaterHfd=ef(Z1f,∙∙∙,Z2df).ThekeypointisthatZsatisfiesHo¨rmander’sconditionatorder2,whichmeansthatthefamily(Z1,∙∙∙,Z2d,[Z1,Zd+1])spansthewholetangentspaceTR2d+1.ForkNandVanopensubsetofHd,wedefinetheassociatedSobolevspaceasfollowingonHk(Hd,V)=fL2(R2d+1)/SuppfVandα/|α|≤k,ZαfL2(R2d+1),whereifα∈{1,∙∙∙,2d}k0,|α|d=efk0andZαd=efZα1∙∙∙Zα0.Asintheclassicalcase,whenksisanyrealnumber,wecandefinethefunctionspaceHs(Hd)throughdualityandcomplexinterpolation,Littlewood-PaleytheoryontheHeisenberggroup(see[6]),orWeyl-Ho¨rmandercalculus(see[10],[12]and[13]).ItturnsoutthatthesespaceshavepropertieswhichlookverymuchliketheonesofusualSobolevspaces,see[4]andtheirreferences.Date:12/03/2006.1
2HAJERBAHOURI,JEAN-YVESCHEMIN,ANDCHAO-JIANGXUThepurposeofthispaperisthestudyoftheproblemsoftraceandtraceliftingonasmoothhypersurfaceofHdintheframeofSobolevspaces.Letuspointoutthattheproblemofexistenceoftraceappearsonlywhensislessthanorequalto1.Indeed,underthesub-sellipicityofsystemZ,thespaceHs(Hd)isincludedlocallyinH2(R2d+1).Soifsisstrictlylargerthan1,thisimpliesthatthetraceonanysmoothhypersurfaceexistsandbelongs1slocallytotheusualSobolevspaceH22ofthehypersurface.Thusthecasewhens=1appearsasthecriticalone.Itisthecasewestudyhere.1.1.Statementoftheresults.Twoverydifferentcasesthenappear:theonewhenthehy-persurfaceisnoncharacteristic,whichmeansthatanypointw0ofthehypersurfaceΣissuchthatZ|w06⊂Tw0Σ,andtheonewhensomepointw0ofthehypersurfaceΣischaracteristic,whichmeansthatZ|w0Tw0Σ.Thenoncharacteristiccaseisnowwellunderstood.In[4],wegiveafullaccountoftraceandtraceliftingresultsonsmoothnoncharacteristichypersurfacesfors>1/2.Thisresultgeneralizevariouspreviousresults(seeamongothers[9],[14]and[23]).LetusrecallthistheoreminthecaseofH1(see[4]forthedetails).Ifw0isanynoncharacteristicpointofΣ,thenthereexistsatlastoneofthevectorfieldsZ1,∙∙∙Z2dwhichistransversetoΣatw0.WedenotebyXΣthesubspaceofTΣdefine,forwinΣ,byXΣ|w=TwΣ∩X|wwhereXistheC-moduleofvectorfieldsspannedby{Z1,∙∙∙,Z2d}.Itiseasilycheckedthat,ifgisalocaldefiningfunctionofΣ,thefamilyfedRj,k=(Zjg)Zk(Zkg)ZjgeneratesXΣandthatitsatisfiestheHo¨rmanderconditionatorder2(seeforinstanceLemma4.1of[4]).Wedefine Hk,ZΣ)=fL2(Σ)/SuppfVΣand(j,k),Rj,kuL2(Σ).Wehaveprovedthefollowingtraceandtraceliftingtheoremin[4].Theorem1.1.LetussupposethatΣisnoncharacteristiconanopensubsetVofHd,thenthetraceoperatoronΣdenotedbyγΣisanontocontinuousmapfromH1(Hd,V)1onto[H1,ZΣ),L2(Σ)]1d=efH2,ZΣ).2RemarkAsthesystemZΣsatisfiestheHo¨rmander’sconditionatorder2,Theorem1.1impliesinparticularthatγΣmapsH1(Hd,V)intoH1/4,VΣ).Weshallnowconsiderthecharacteristiccase.ThesetofcharacteristicpointsofΣnΣc=wΣ/Z|wTwΣ},mayhaveacomplicatedstructure.Forthesakeofsimplicity,weshallonlyconsiderhereaparticularcase.BytranslationintheHeisenberggroup,weworkonlyinaneighborhoodofw0=(0,0,0).Nearw0,thehypersurfaceΣcanalwaysbewrittenasΣ={w=(x,y,s)/g(w)d=efwf(x,y)=0}withf(0,0)=0andDf(0,0)=0.Fromnowon,weassumethatfisahomogenenouspolynomialofdegree2onR2d.Inthiscase,theequationishomogenenousoforder2withrespecttothedilationofHeisenberggroupdλ(x,y,s)d=ef(λx,λy,λ2s).ThenthesetofcharacteristicpointsΣcisasubmanifolddefinedbyΣc={w=(s,x,y)Hd/g(w)=0andhLj,(x,y)i=0}whereLjisthelinearform(onR2d)definedbyZj(g)(x,y).
TRACETHEOREMONTHEHEISENBERGGROUP3Letusdenotebyrtherankofthefamily(Lj)1j2doflinearformsonR2d.Letusobservethatrisalsotherankofthematrix(ZiZjg)1i,j2datw0.Letusnoticethat,because,ifi∈{1,∙∙∙d}andj6=i+d,(ZiZi+dg)(w0)(Zi+dZig)(w0)=2sg(w0)=2and(ZiZjg)=(ZjZig),therankofthematrix(ZiZjg)1i,j2dandthusof(Lj)1j2disatleastd.Fromnowon,wealwaysconsiderthiscaseforthesakeofsimplicity.Letusintroducesomeringsoffunctionsadaptedtooursituation.Definition1.1.LetWbeanyopensubsetofΣandFaclosedsubsetofW.LetusdenotebyCF(W)thesetofsmoothfunctionsaonW\Fsuchthatforanymulti-indexα,aconstantCαexistssuchthatαNd|αa(z)|≤Cαd(z,F)−|α|,whereddenotesthedistanceonΣinducedbytheeuclidiandistanceonR2d+1.NowletusdefinethevectorfieldsonΣwhichwilldescribetheregularityonΣ.Definition1.2.LetWbeaneighhourhoodofw0.WedenotebyZΣtheCΣc(W)modulusspannedbythesetvectorfieldsofZ∩TΣ|WthatvanishonΣc.AsweshallseeinProposition3.1,themodulusZΣisoffinitetype(ofcourseasaCΣc(W)modulus)ifw0isaregularcharacteristicpointandWischoosensmallenough.IfgisalocaldefiningfunctionofΣ,ageneratingsystemisgivenbyfedRj,k=(Zjg)Zk(Zkg)Zjfor1j<k2d.(1.1)Nowwearereadytointroducethespaceoftraces.Definition1.3.LetWbeasmallenoughneighbourhoodofw0.WedenotebyH1(ZΣ,W)thespaceoffunctionsvofL2(Σ)supportedinWsuchthatX2def22kvkH1(ZΣ)=kvkL2(Σ)+kRj,kvkL2(Σ)<.1j,k2dswherethefamily(Rj,k)1j,k2disgivenby(1.1).Ifs[0,1],wedefineH(ZΣ,V)bycomplexinterpolation.Ourtheoremisthefollowing.Theorem1.2.LetVbeasmallenoughneighhourhoodofw0.ThentherestrictionmapγΣ1isanontocontinuousmapfromH1(Hd,V)ontoH2(ZΣ,VΣ).Letusremarkthat,ifr=2d,thistheoremisaparticularcaseofTheorem1.8of[4].1.2.Structureoftheproof.Inourpaper[4],whichconrresponedstothecasewhenr=2dasthusΣc={w0},weuseablowupofthepointw0.HereweblowupthesubmanifoldΣc.Inordertodoit,letusintroduceafunctionϕ∈D(R+\{0})suchthatXt[1,1]\{0},ϕ(2pt)=1.(1.2)0=pdef1Letusdefinethefunctionρcbyρc=g2+|rHg|44.NowwritingthatforanyfunctionuinL2(ρc1),fedXu=ϕpuwithϕp(w)=ϕ(2pρc(w)),(1.3)0=p
4HAJERBAHOURI,JEAN-YVESCHEMIN,ANDCHAO-JIANGXUweapplyTheorem1.1oftraceandtraceliftingtoeachpieceϕpuwhichissupportedinadomainwhereΣisnoncharacteristicbecauseρc2pinthisdomain.ThisdecompositionleadsimmediatelytotheproblemofestimatingthenormH1(Hd)ofeachpieceϕpu.LeibnitzformulaandthechainruletellusthatrH(ϕpu)=ϕprHu+2pϕ0(2pρc)urHρc.Letusobservethat,asd2X24Zjρc=2g(Zjg)+4|rHg|(Zkg)Zj(Zkg),1=kwehave,foranyrealnumbers,|rHρcs|≤Csρcs1.Asthesupportofϕ0(2pρc)included0inρc2p,thesupportsofϕ0(2pρc)andϕ0(2pρc)aredisjointif|pp0|≥N0forsomeN0.Thus,wegetthatX u 222pkϕ0(2pρc)urHρck22C .L2p=0ρcLThisleadstotheproofofthefollowingHardytypeinequality.Theorem1.3.AneighbourhoodVofw0existssuchthat,foranyuinthespaceH1(Hd,V)ofH1(Hd)functionssupportedinV,Zu212dwCkrHuk2L2.withρc=g2+|rHg|44.ρdcHThistheoremimpliesthat,foranyuinH1(Hd,V),X22krH(ϕpu)kL2CkrHukL2.(1.4)0=pTheproofofthistheorem,whichisthecoreofthiswork,isthepurposeofthesecondsection.Inthethirdsection,afterdilation,weapplyTheorem1.1.Thisgivesaratherunpleasantdescriptiononthetracespace.Then,weuseaninterpolationresultwhichallowstoconcludetheproofofTheorem1.2.2.AHardytypeinequality2.1.TheclassicalHardyinequality.Asawarmup,letusrecallbrieflytheusualproofoftheclassicalHardyinequality1.Zu21dwCkrHuk22withρ(w)=s2+(|x|2+|y|2)24.(2.5)L2ρdHAsD(Hd\{0})isdenseinH1(Hd),wehaverestrictourselvestofunctionsuinD(Hd\{0}).ThentheproofmainelyconsistsinanintegrationbypartswithrespecttotheradialvectorfieldRHadaptedtothestructureofHd,namelydefXdXdRH=2s∂s+xjxj+yjyj=s[Y1,X1]+(xjXj+yjYj)j=1j=11ForadifferentapproachbasedonFourieranalysis,see[2].
TRACETHEOREMONTHEHEISENBERGGROUP5oncenoticedthatRHρ2=2ρ2anddivRH=2d+2.Moreprecisely,thisgivesZu2ZXduxjyjZsZsd2dw=Xj+YjudwY12u(X1u)dw+X12u(Y1u)dw.ρj=1ρρρρρs  Aswehave Zj 1,Cauchy-Schwarzinequalitygives(2.5).2ρ2.2.ConstructionofsubstituteofρandRH.Theclassicalcasestudiedabovecorre-spondstothecasewhenr=2d.Letusassumefromnowonthatr<2dandletusconsider(Lj`)1`rabasisofthevectorspacegeneratedby(Lj)1j2d.First,wehavethefollowinglemma.Lemma2.1.Acoupleofvectorfields(Z0,Z0)existsin(Z\{Zj1,∙∙∙,Zjr})×(±Z)suchtaht[Z0,Z0]=2sandD(Z0g)(w0)6=0.ProofofLemma2.1LetusconsiderZ0∈Z\{Zj1,∙∙∙,Zjr}.andZ0in±Zsuchthat[Z0,Z0]=2s.If±Z0belongsto{Zj1,∙∙∙,Zjr},weinferfromthedefinitionofthefamily(Lj`)1`rthatD(Z0g)(w0)isdifferentfrom0andthenZ0=Z0fits.If±Z0isnotin{Zj1,∙∙∙,Zjr},as(Z0(Z0g))(w0)(Z0(Z0g))(w0)=2,eitherD(Z0g)(w0)orD(Z0g)(w0)isdifferentfrom0.ThusifD(Z0g)(w0)=0,wegetthelemmainterchangingtheroleofZ0andZ0.Letusstatethefollowingtheorem,whichimmediatlyimpliestheHardytypeinequalitystatedinTheorem1.3.Theorem2.1.AneighbourhoodVofw0existssuchthat,foranyuinH1(Hd,V),Zu2def12dwCkrHuk2L2withρ0=g2+(Z0g)44.ρ0TheaboveinequalityisobviouslybetterthantheoneofTheorem1.3anditissurprisinglytheoneweareabletoprove.ProofofTheorem2.1Bydefinitionofthefamily(Lj)1`r,afamilyofrealnumbers(α`)1`rexistssuchthatrXZ0g=α`(Zj`g).(2.6)1=`LetusdefineananalogoustoRHisoursituationby1defrXR1=2g∂s+(Z0g)Ze0withZe0=Z0α`Zj`.(2.7)21=`InordertocheckthatR1isanalogoustotheradialfieldinthecaseoftheclassicalHardyinequality,letusprovethatR1ρ04=4ρ04anddivR1=3.(2.8)Bydefinitionofthefunctionρ0,wehaveR1ρ04=2g(R1g)+4(Z0g)3R1(Z0g).
6HAJERBAHOURI,JEAN-YVESCHEMIN,ANDCHAO-JIANGXUInequality(2.6)impliesthatZe0istangenttoΣ.Usingthatsg1,wegetthatR1g=2g.LetuscomputeR1(Z0g).Ass(Z0g)=0,wehave1R1(Z0g)=(Z0g)Ze0(Z0g).2ThenwehaveR1ρ04=4ρ04.LetusnoticethatZ0doesnotbelongtothefamily(Zj`)1`r.ThusZ0commuteswiththevectorfieldsZj`.BydefinitionofZe0,weinferrX[Ze0,Z0]=[Z0,Z0]+α`[Z`,Z0]=2s.(2.9)1=`BydefinitionofZe0,wehaveZe0g=0.ThuswegetZe0(Z0g)=Z0(Ze0g)+2sg=2.(2.10)ItturnsoutthatR1ρ14=4g2+4(Z0g)4=4ρ04.Now,letuscomputedivR1.Wehave1divR1=2sg+Ze0(Z0g)+(Z0g)divZe0.2UsingthatthevectorfieldsZjaredivergencefree,thefactthatsg1and(2.10),wegetthatdivR1=3.ThusAssertion(2.8)isproved.InordertocontinuetheproofofTheorem2.1,letusobservethat,nearw0,thesetρ01(0)isasubmanifoldofHdofcodimension2.ThefollowinglemmawillallowustoassumeallalongtheproofthatubelongstoD(V\ρ01(0)).Lemma2.2.LetVbeaboundeddomainofHdandΓasubmanifoldofcodimension2.ThenD(V\Γ)isdenseinthespaceH01(Hd,V)offunctionsofH01(Hd)supportedinVequippedwiththenorm1kuk2L2+krHuk2L22.ProofofLemma2.2AsH01(Hd,V)isaHilbertspace,itisenoughtoprovethattheorthogonalofD(V\Γ)is{0}.Letubeinthisspace.ForanyvinD(V\Γ),wehave(u|v)L2+(rHu|rHv)L2=0.Byintegrationbypart,thisimpliesthatv∈D(V\Γ),huΔHu,vi=0.ThusthesupportofuΔHuisincludedinΓ.AsZjubelongstoL2,thenZj2ubelongstoH1(R2d+1)(theclassicalSobolevspace).Andexcept0,nodistributionofH1(R2d+1)canbesupportedinasubmanifoldofcodimensiongreaterthan1.ThusuΔHu=0.TakingtheL2scalarproductwithuimpliesthatu0.ThankstoInequality(2.7),wehave1ρ2=R1ρ2.(2.11)002Thusbyintegrationbypart,wehave,usingInequality(2.7),ZZZu23u2defu2dw=2dw+IwithI=2(R1u)dw.ρ02ρ0ρ0
TRACETHEOREMONTHEHEISENBERGGROUP7InordertoestimateI,whichcontainstermsofthetypeg∂su,wehavetocomputethevectorfieldR1intermofelementsofZ.Using(2.9),weinferthatR1=2g[Ze0,Z0]+1(Z0g)Ze0.2WededucethatI=J1+J2withZJ1d=ef1u(Z0g)(Ze0u)dwandZ2ρ0ρ0ufedJ2=2g[Ze0,Z0]udw.ρ0Bydefinitionofρ0,CauchySchwarzinequ alit yyieldsu |J1|≤C ρ L2krHukL2.(2.12)0TheestimateaboutJ2isalittlebitmoredifficulttoobtain.LetuswritethatJ2=K1K2htiwdefZudefZuK1=2gZe0(Z0u)dwandK2=2gZ0(Ze0u)dw.ρρ00Byintegrationbyparts,wehaveK1=K11K12withZgfedK11=ρ2(Ze0u)(Z0u)dwand0ZK12d=effu(Z0u)dwwithfd=efρ0Ze0g2ρρ00Bydefinitionofρ0,itisobviousthat|K11|≤CkrHuk2L2.UsingthatZe0g=0,weget Ze0g2 =2g6 Ze0(Z0g) |Z0g|3Cg3Cρ0ρ0ρ0ρ0ThisensuresthatfisboundedonVandthusbyCauchy-Schwarzinequality,u kK12k≤C 2krHukL2.ρL0Togetherwith(2.13),thisprovesthatu |K1|≤C ρ L2+krHukL2krHukL2.0InordertoestimateK2,letuswritethat,byintegrationbyparts,ZgZguK2=ρ2(Z0u)(Ze0u)dw+ρ0Z0ρ2ρ(Ze0u)dw.000Usingthat�Z0ρ04=2g(Z0g)+4Z0(Z0g)(Z0g)3,gweimmediatlygetthatthefunctionρ0Z02isboundedonVandwededucethatρ0u |K2|≤C ρ0 L2+krHukL2krHukL2.(2.13)(2.14)
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