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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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