support theorems for the light ray transform on analytic lorentzian manifolds plamen stefanov资料.pdf

该【support theorems for the light ray transform on analytic lorentzian manifolds plamen stefanov资料 】是由【李十儿】上传分享，文档一共【16】页，该文档可以免费在线阅读，需要了解更多关于【support theorems for the light ray transform on analytic lorentzian manifolds plamen stefanov资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..PROCEEDINGSOFTHEAMERICANMATHEMATICALSOCIETYVolume145,Number3,March2017,Pages1259–1274http://dx./,2016SUPPORTTHEOREMSFORTHELIGHTRAYTRANSFORMONANALYTICLORENTZIANMANIFOLDSPLAMENSTEFANOV(CommunicatedbyMichaelHitrik)(?,+,...,+)onthemanifoldMofdimension1+n,n≥(s)(alsocallednullgeodesics)arethesolutionsofthegeodesicequation?sγ˙=0forwhichg(˙γ,γ˙)=?xedchoiceofit,heweightedlightraytransformLκfofafunction(oradistribution)fonMby()Lκf(γ)=κ(γ(s),γ˙(s))f(γ(s))ds,,positivelyho-mogeneousinitssecondvariableofdegreezero,=1,?nitionofthegeodesicswillbespeci?edbelowbutinallcases,;see,.,[3,24–26,31,37,38]fortimedependentcoe?cients,andalso[2,22]-plexofcurvesisrestrictedtothelowerdimensionmanifoldg(˙γ,γ˙)=,includingitsmicrolocalinvertibil-(C∞)spacelikewavefrontsetoff,[31],theauthorshowedthatifgistheMinkowskimetric,andiff(t,x)issupportedinacylinderR×B(0,R)andhastemperedgrowthinthetimevariable,thenLfdeterminesfuniquely;seealso[25].TheproofwasbasedonthefactthatLfrecoverstheFouriertransformf?off()inthespacelikecone|τ|<|ξ|inadirect(andstable)wayandsincef?(τ,ξ)isanalyticintheξvariable(withvaluedistributionsintheτvari-able),,2015and,inrevisedform,January18,?-1301646.:13:;seernal-terms-of-use:..?inthetimelikecone|τ|>|ξ|(truealsointhemostgeneralLorentziancase;seethenextparagraph)thusLhasahighdegreeofinstability;seealso[1].Fromaphysicalpointofview,thiscouldbeexpected:wecanrecoverall“signals”movingslowerthanlight,andweshouldnotexpecttorecoverthosemovingfasterthanlight;?at,itisfairlyobviousthatLκfcannot“see”thewavefrontWF(f)inthetimelikecone,thisjustfollowsfromtheinspectionofthewavefrontoftheSchwartzkernelofLκ;(f)inthespacelikeconeisfarlessobviousandcertainlyrequiressomegeometricassumptionslikenoconjugatepointsorexistenceofafoliationofstrictlyconvexhypersurfaces,Lκasin[8,9,11].ThatoperatorisintheIp,lclassκofΨDOswithsingularkernels,whichareFourierIntegralOperators(FIOs),infact;see[10]LκintheMinkowskiκcaseforn=2ispresentedin[8,9,11],lcalculustogetmorere?nedmicrolocalresults,however,requirestheconeconditionwhichcannotbeexpectedtoholdongeneralRiemannianmanifoldsduetothelackofsymmetry,aspointedoutin[11].AnalternativeapproachtorecovertheC∞spacelikesingularitiescanbefoundin[19].OurmainresultissupporttheoremsandinjectivityofLκforanalyticmetricsandweights(onanalyticmanifoldsM).ItcanbeviewedasanextensionoftheclassicalHelgasonsupporttheoremforRadontransformsintheEuclideanspace[14].[4–6].In[5],theauthorsprovesupporttheoremsforRadontransforms(with?atgeometry)[6],theystudy“plexes”inR1+2withanalyticweights,,however,arebasedonthecalculusoftheanalyticFIOsasananalyticversionoftheC∞analysisin[9].Suchageneralizationdoesnotexisttothebestoftheauthor’(see,.,[35]),andananalyticversionoftheFIOcalculus,includingtheIp,lone,[17,18]andrelatedresults,evenfortensor?eldsin[7,33,34].AbreakthroughwasmadebyUhlmannandVasyin[36],wheretheyprovedasupporttheoremintheRiemanniancasenearastrictlyconvexpointofahypersurfaceindimensionsn≥-raytransformisassumedtobezeroonallgeodesicsclosetotangentonestothehypersurfaceatthatpoint,[20]andthen≥?cultiesrelatedtothesingularitiesofthesymbolofLLκ:weformsmoothtimelikehypersurfacesfoliatedκputeaweightedRadontransformRbyjustapplyingFubini’(non-restricted)Radontransformknownonanopensetofhypersurfaces,whichinthesmoothcaseisdoablewithclassicalmicrolocaltechniquesgoingbacktoGuillemin[12,13].:13:;seernal-terms-of-use:..SUPPORTTHEOREMSFORTHELIGHTRAYTRANSFORM1261transformaswell[15].Ontheotherhand,thisapproachdoesnotallowustoanalyzethelightlikesingularities,wheresomeformoftheIp,RasananalyticΨDObutitisnotclearhowtodothattoobtainpurelylocalresultsduetothedelicatenatureofcut-o?,weusetheanalyticstationaryphaseapproachbySj¨ostrand[28]alreadyusedbytheauthorandUhlmannin[34];seealso[7,17,18].AsasimpleexampleillustratingthereductionoftherestrictedraytransformLκtoaclassicalRadontransformR,(1,θ),with|θ|=(withanormalν=(νt,νx)suchthat|νt|<|νx|)∈C∞(oranalytic),,;,wenoticethatsomeglobalconditionsonthegeodesic?owareclearlyneededformicrolocalinversion,,ifg=?dt2+hij(x)dxidxj,wherehisaRiemannianmetriconaboundeddomain,thenLκ,restrictedtoat-independentfunction,reducestothegeodesicX-[21,30]thatwhenn=2,XfrecoversWF(f)≥3,thenoconjugatepointsconditionissu?cient[33,34]andthereareexamplesofmetricsofproducttypeforwhichitisnecessary,bythe2Dresultsin[21,30].Ontheotherhand,thesupporttheoremin[36]providesglobaluniquenessandstabilityunderanothertypeofcondition:existenceofafoliationbystrictlyconvexhypersurfaces(conjugatepointsmayexist).Thisimpliesstableinvertibilitywhenn≥,incontrastto[36],heren=2isallowedsinceforourpurposesfullellipticity(inalldirectionsatapoint)isnotneeded;onlyellipticityatdirectionsconormaltothefoliationsu?,,=?dt2+(dx1)2+···+(dxn)2betheMinkowskimetricinR1+→(t+s,x+sθ)with|θ|=“unit”speedisnotinvariantlyde?nedunderLorentziantransformationsbut,ina?xedcoordinatesystem,thescalingparameter1(.,dt/ds=1)()Lf(x,θ)=f(s,x+sθ)ds,x∈Rn,θ∈Sn?:13:;seernal-terms-of-use:..?nitionisbasedonparameterizationofthelightlikegeodesics(lines)bytheirpointofintersectionwitht=0anddirection(1,θ).Wewillusethenotation() x,θ(s)=(s,x+sθ).Theparameterization(x,θ)de?∈C∞(R×Rn×Sn?1),heweightedversionLκofLbyLκf(x,θ)=κ(s,x+sθ,θ)f(s,x+sθ)ds,x∈Rn,θ∈Sn?,vectorsv=(v0,v)satisfying|v0|<|v|(.,g(v,v)>0)(0,v),v=|v0|>|v|(.,g(v,v)<0)aretimelike;anexampleis(1,0):g(v,v)=,thede?nitionisthesamebutwereplacegbyg?1,,respectivelyspacelike,ifitsnormal(whichisacovector)isspacelike,?()K?{(t,x);|x|≤c|t|+R}forsome0<c<1,R>(),-=?Kissmooth,,∈D(R1+n) x0,θ0bea?xedlightlikelineintheMinkowskispacetimeandletU(x,θ)beanopenandconnectedsubsetofRn×Sn?(t,x,θ)be00analyticandnon-vanishingfor(t,x)nearsuppfsothat(x?tθ,θ)∈(x,θ)=0inUandif x0,θ0doesnotintersectsuppf,thennoneofthelines x,θ,(x,θ)∈U,(M,g)(null)geodesicsarede?nedasthegeodesicsγ(s)forwhichg(˙γ,γ˙)=(s)=as+b,a=0,γ?,?xedlightlikegeodesicγ0(s),intersectingSfors=(0)∈Sclosetoγ0withdirectionscloseto˙γ0(0)withinitialpointsxonSandinitiallightlikedirectionsvatxpointinginthedirectionofγ˙(s)canbe?xedbyrequiringγ(0)∈ponentof˙γonStobeagivennegativefunction,forexample,?:13:;seernal-terms-of-use:..SUPPORTTHEOREMSFORTHELIGHTRAYTRANSFORM1263smooth/analytic,wecalltheparameterizationsmooth/?nesatopologyandasmooth/analyticstructureofthelightlikegeodesicsde?nedona??Misclosed,wecallthenullgeodesicγ(s)non-trappinginC,andifγ?1(C)iscontainedinsomeopen?niteinterval,,themaximallyextendednullgeodesicwit