Differentiability of Lipschitz maps from metric me

arXiv:0808.3249v1[math.MG]24Aug2008DIFFERENTIABILITYOFLIPSCHITZMAPSFROMMETRICMEASURESPACESTOBANACHSPACESWITHTHERADONNIKODYMPROPERTYJEFFCHEEGERANDBRUCEKLEINERAbstract.InthispaperweprovethedifferentiabilityofLips-chitzmapsX→V,whereXisacompletemetricmeasurespacesatisfyingadoublingconditionandaPoincar´einequality,andVdenotesaBanachspacewiththeRadonNikodymProperty(RNP).Theproofdependsonanewcharacterizationofthedifferentiablestructureonsuchmetricmeasurespaces,intermsofdirectionalderivativesinthedirectionoftangentvectorstosuitablerectifi-ablecurves.Contents1.Introduction12.InverselimitsandtheANP53.Weakderivatives64.Velocitiesofcurves85.Anewcharacterizationoftheminimaluppergradient13Appendix.TheANPimpliestheRNP(AnalternateproofofthetheoremofJames-Ho[JH81])15References161.IntroductionInthispaperwewillusethetermPIspacetorefertoaλ-quasi-convexcompletemetricmeasurespace(X,dX,μ)satisfyingadoublingDate:August24,2008.ThefirstauthorwaspartiallysupportedbyNSFGrantDMS0105128andthesecondbyNSFGrantDMS0701515.12JEFFCHEEGERANDBRUCEKLEINERcondition(1.1)μ(B2r(x))≤2κ·μ(Br(x)),andp-Poincar´einequality,(1.2)−ZBr(x)|f−fx,r|dμ≤τr−ZBλr(x)gpdμ1p,wherex∈X,r∈(0,∞),fisacontinuousfunction,gisanuppergradientforf,−ZAfdμ:=1μ(A)ZAfdμ,fx,r:=−ZBr(x)fdμ;see[HK96,Che99,Hei01].WewillalsoassumethatthecollectionofmeasurablesetsisthecompletionoftheBorelσ-algebrawithrespecttoμ(i.e.everysubsetofasetofmeasurezeroismeasurable)[Roy88,p.221].Sometimes,byabuseoflanguage,wejustsaythatXisaPIspace.Thenotationabove,inparticularthespaceX,themeasureμ,aswellastheconstantsκandλ,willbemaintainedthroughoutthepaper.In[Che99],adifferentiationtheoryforrealvaluedLipschitzfunc-tionsonPIspaceswasgiven.Thenotionofdifferentiationisex-pressedintermsofanatlas.Anatlasconsistsofacountablecol-lection{(Uα,yα)}α∈Aofcharts,wheretheUα’saremeasurablesubsets,μ(X\Sα∈AUα)=0,yα:X→Rk(α)isLipschitz,andthechartssatisfycertainadditionalconditions.Weputyα=(yα1,···,yαk(α)).LetVdenoteaBanachspace.Definition1.3.ALipschitzmapf:X→Visdifferentiablealmosteverywherewithrespecttotheatlas{(Uα,yα)}α∈Aifthereisacollec-tion∂f∂yαm:Uα−→Vα∈A,1≤m≤k(α)ofBorelmeasurablefunctionsuniquelydeterminedμ-almostevery-where,suchthatforalmosteveryx∈Uα,(1.4)f(x)=f(x)+k(α)Xm=1∂f∂yαm(x)(yαm(x)−yαm(x))+o(dX(x,x)).DIFFERENTIABILITYOFRNPVALUEDLIPSCHITZMAPS3Wewillsaythatfisdifferentiableataspecificpointx∈Xif(1.4)holdsforthatpoint.ThecaseV=RofDefinition1.3wasconsideredin[Che99];oneofthemainresultsthere[Che99,Theorem4.38]wastheexistenceofanatlaswithrespecttowhicheveryLipschitzfunctionf:X→Risdifferentiablealmosteverywhere.Wewillfixsuchanatlasthroughoutthepaper.Itfollowsreadilyfromthedefinitionsthatif(Uα,yα)and(¯Uα,¯y¯α)arechartsfromtwosuchatlases,thenthematrixofpartialderivatives∂yαm∂¯y¯α¯misdefinedandinvertiblealmosteverywhereintheover-lapUα∩¯U¯α.Thisalsoyieldsbi-LipschitzinvariantmeasurabletangentbundleTX.Themainresultofthispaperis:Theorem1.5.EveryLipschitzmapfromXintoaBanachspacewiththeRadon-NikodymPropertyisdifferentiableμ-almosteverywhere.WerecallthataBanachspaceVhastheRadon-NikodymProperty(RNP)ifeveryLipschitzmapf:R→VisdifferentiablealmosteverywherewithrespecttoLebesguemeasure.SinceRisanexampleofaPIspace,Theorem1.5isoptimalinthesensethattheclassofBanachspacetargetsconsideredismaximal.Justasin[Che99],[CK06],thedifferentiationtheoremaboveimposesstrongrestrictionsonPIspaceswhichbi-LipschitzembedinRNPtar-gets,andmaythereforebeusedtodeducenonembeddingtheorems.Theorem1.6.IfXadmitsabi-LipschitzembeddinginaBanachspacewiththeRNP,thenforμ-a.e.x∈X,everytangentconeatxisbi-LipschitzhomeomorphictoaEuclideanspace.Discussionoftheproof.TheproofofTheorem1.5exploitstheframeworkintroducedin[CK06],whichinvolvesinversesystemsoffinitedimensionalBanachspacesandtheirinverselimits;seeSection2fortherelevantdefinitions.Itwasobservedin[CK06]thatanyseparableBanachspaceVcanberealizedasasubspaceofaninverselimitspaceV⊂lim←−Wi,wherelim←−WiistheinverselimitofaninversesystemoffinitedimensionalBanachspaces.Anadvantageofthisviewpointisthatthedifferentia-bilitytheoryforrealvaluedLipschitzfunctionsleadsimmediatelytoanaturalnotionofaweakderivativeofLipschitzmapf:X→V,whichisamaptakingvaluesinlim←−Wi.4JEFFCHEEGERANDBRUCEKLEINERItwaspointedoutin[CK08]thatinverselimitslim←−WiofinversesystemsoffinitedimensionalBanachspacesarepreciselythedualsofseparableBanachspaces.SinceVhastheRNP,byaresultofGhoussoub-Maurey[GM84],onecanchooseanembeddingV⊂lim←−Wiasabove,sothatthepair(lim←−Wi,V)hastheAsymptoticNormingProperty(ANP).TheANPwasintroducedbyJames-Ho[JH81],whoshowedthatitimpliestheRNP,compareSection3andtheappendix.ThefirststepintheproofofTheorem1.5istoshowthatiftheweakderivativeofftakesvaluesinthesubspaceV⊂lim←−Wi,thenfisdifferentiableμ-a.e.Theargumentforthisisbrief,andillustratesthesmoothinteractionbetweentheinverselimitsetup,theANP,andbasictheoremsofmeasuretheory(Egoroff’sandLusin’stheorems).Asanotherillustrationofthissmoothinteraction,intheappendixwegiveashortproofoftheJames-Hotheorem[JH81].Theremainderoftheproof,whichappearsinSection4,isdevotedtoprovingthattheweakderivativeofftakesvaluesinV.Aheurist