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TRACE THEOREM ON THE HEISENBERG GROUP ON HOMOGENEOUS HYPERSURFACES

12 pages
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 ·


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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
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