An inverse function theorem for Fréchet spaces satisfying a smoothing property and (DN)

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1、Math.Nachr.206(1999),123-145AnInverseFunctionTheoremforF’rechetSpacesSatisfyingaSmoothingPropertyand(DN)BYMARKUSPOPPENBERGofDortmund(ReceivedJanuary14,1997)Abstract.ClassicalinversefunctiontheoremsofNash-MosertypeareprovedforFMchetspacesthatadmitsmoothingoperatorsasintroducedby

2、NASH.InthisnoteaninversefunctiontheoremisprovedforF’rCchetspaceswhichonlyhavetosatisfythecondition(DN)ofVOGTandthesmoothingproperty(Sn),;forinstance,anyFrhchet-Hilbertspacewhichisan(a)-spaceinstandardformhasproperty(Sn)t.ThemainresultofthispapergeneralizesatheoremofLOJASIEWICZa

3、ndZEHNDER.ItcanbeappliedtothespaceCm(K)ifthecompactKCIRNistheclosureofitsinteriorandsubanalytic;differentfromclassicalresultstheboundaryofKmayhavesingularitieslikecusps.Thegrowthassumptionsonthemappingsareformulatedintermsoftheweightedmultiseminorms[Im,l;introducedinthispaper;n

4、onlinearsmoothpartialdifferentialoperatorsonCm(K)andtheirderivativessatisfytheseformalassumptions.0.IntroductionClassicalinversefunctiontheoremsofsocalledNash-MosertypeareprovedforF’rCchetspacesthatadmitsmoothingoperators(cf.[2],[3],[5],[8],[16],[18]).Thesesmoothingoperatorshav

5、ebeenintroducedbyNASH[9]toovercomeadifficultywhichMOSER[8]calledtheproblemoflossofderivatives.Theassumptionofsmoothingoperatorsisrestrictive;eachnuclearFrCchetspaceadmittingsmoothingoperatorsis(tamely)isomorphictosomepowerseriesspaceofinfinitetype(cf.VOGT[21]).Thepurposeofthisp

6、aperistoproveaninversefunctiontheoremforamoregeneralclassofF’rCchetspacesgeneralizingaresultofLOJASIEWICZ,ZEHNDER[5].Inplaceofsmoothingoperatorsonlythetopologicalcondition(DN)ofVOGT[20]andsmoothingproperty(Sn),(cf.(111)aresupposed.Forexample,thespaceC”(K)hasbothpropertiesifthec

7、ompactKcENistheclosureofitsinteriorandsubanalytic([l]).1991MathematicsSubjectClassification.Primary:58C15;Secondary:46A04.Keywordsandphrases.Inversefunctiontheorem,implicitfunctiontheorem,Nash-Moser,partialdifferentialoperator,F’rCchetspace,tamelinearmap,nonsmoothdomains,subana

8、lytic.124Math.Nachr.206(19Differentfromknownresultsthe