Rellich’s lemma for Sobolev spaces .pdf

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1、68BAN-CRAINIC,ANALYSISONMANIFOLDSThus,bydensityofCc∞(Rn),theinducedmapβ1:Hs(Rn)→H¯−s(Rn)∗isanisometry.Likewise,β2:H−s(Rn)→H¯s(Rn)∗isanisometry.Fromtheinjectivityofβ1itfollowsthatβ2hasdenseimage.Beinganisometry,β2mustthenbesurjective.Likewise,β1issurjective.¤4.5.Rellich’slemmaforSobolevspacesInthi

2、ssectionwewillgiveaproofoftheRellichlemmaforSobolevspaces,whichwillplayacrucialroleintheproofoftheFredholmpropertyforellipticpseudo-differentialoperatorsoncompactmanifolds.Givens∈RandacompactsubsetK⊂Rn,wedefineH(Rn)={u∈H(Rn)

3、suppu⊂K}.s,KsLemma4.5.1.Hs,K(Rn)isaclosedsubspaceofHs(Rn).ProofLetf∈Cc∞(Rn

4、).Thenthespacef⊥:={u∈H(Rn)

5、hu,fi=0}shasFouriertransformequaltothespaceofϕ∈L2s(Rn)withhϕ,Ffi=0,whichistheorthocomplementof(1+kξk)−2sFfinL2s(Rn).AsthisorthocomplementisclosedinL2s(Rn),itfollowsthatf⊥isclosedinHs(Rn).WenowobservethatHs,K(Rn)istheintersectionofthespacesf⊥forf∈Cc∞(Rn)withsuppf∩K=∅.¤Le

6、mma4.5.2.(Rellich)Lett

7、hetspaceC(Rn).ProofBoundednessofBmeansthateverycontinuoussemi-normofC1(Rn)isboundedonB.LetK⊂Rnbeacompactball.ThenthereexistsaconstantC>0suchthatsupKkdfk≤Cforallf∈Bandeach1≤j≤n.SinceZ1f(x)−f(y)=df(y+t(x−y))(x−y)dt0forally∈x,weseethat

8、f(y)−f(x)

9、≤Ckx−yk,forall(x,y∈K).ItfollowsthatthesetoffunctionsB

10、

11、K={f

12、K

13、f∈B}isequicontinuousandboundedinC(K).ByapplicationoftheAscoli-Arz`elatheorem,thesetB

14、KisrelativelycompactinC(K).Inparticular,if(fk)isasequenceinB,thenthereisasubsequence(fkj)whichconvergesuniformlyonK.Let(fk)beasequenceinB.Weshallnowapplytheusualdiagonalproce-duretoobtainasubsequencethatco

15、nvergesinC(Rn).LECTURE4.FOURIERTRANSFORM69Forr∈NletKrdenotetheballofcenter0andradiusrinRn.Thenbyrepeatedapplicationoftheabovethereexistsasequenceofsubsequences(fk1,j)º(fk2,j)º···...suchthat(fkr,j)convergesuniformlyonKr