On the Asymptotic Completeness of the Volterra Cal

arXiv:math/0310184v5[math.AP]30Aug2004ONTHEASYMPTOTICCOMPLETENESSOFTHEVOLTERRACALCULUSRAPHA¨ELPONGEWITHANAPPENDIXBYH.MIKAYELYANANDR.PONGEAbstract.TheVolterracalculusisasimpleandpowerfulpseudodiffer-entialtoolforinvertingparabolicequationsandithasalsofoundmanyapplicationsingeometricanalysis.Ontheotherhand,animportantpropertyinthetheoryofpseudodifferentialoperatorsistheasymptoticcompleteness,whichallowsustoconstructparametricesmodulosmooth-ingoperators.InthispaperwepresentnewandfairlyelementaryproofstheasymptoticcompletenessoftheVolterracalculus.IntroductionThispaperdealswiththeasymptoticcompletenessoftheVolterracal-culus.Recallthatthelatterwasinventedintheearly70’sbyPiriou[Pi1]andGreiner[Gr]andconsistsinamodificationoftheclassicalΨDOcalculusinordertotakeintoaccounttwoclassicalpropertiesoccurringinthecon-textofparabolicequations:theVolterrapropertyandtheanisotropywithrespecttothetimevariable(cf.Section1).AsaconsequencetheVolterracalculusprovedtobeapowerfultoolforinvertingparabolicequations(seePiriou([Pi1],[Pi2]))andforderivingsmallheatkernelasymptoticsforellipticoperators(seeGreiner[Gr]).Subsequently,theVolterracalculushasbeenextendedtoseveralothersettings.In[BGS]Beals-Greiner-StantonproducedaversionoftheVolterracalculusforthehypoellipticcalculusonHeisenbergmanifolds([BG],[Ta])andusedittoderivethesmalltimeheatkernelasymptoticsfortheKohnLaplacianonCRmanifolds.Also,Melrose[Me]fittheVolterracalculusintotheframeworkofhisb-calculusonmanifoldswithboundaryandusedittoinverttheheatequationwiththepurposeofproducingaheatkernelproofoftheAtiyah-Patodi-Singerindextheorem[APS].Morerecently,Buchholz-Schulze[BS],Krainer([Kr1],[Kr2])andKrainer-Schulze[KS]extendedtheVolterracalculustothesettingoftheconecalculus2000MathematicsSubjectClassification.Primary35S05.Keywordsandphrases.Pseudodifferentialoperators,Volterracalculus.TheauthorwaspartiallysupportedbytheEuropeanRTNetworkGeometricAnalysisHPCRN-CT-1999-00118.1ofSchulze([Sc1],[Sc2])inordertosolvegeneralparabolicproblemsonman-ifoldwithconicalsingularitiesandtodealwithlargetimeasymptoticsofsolutionstoparabolicproblemsonmanifoldswithboundary(bylookingattheinfinitetimecylinderasamanifoldwithaconicalsingularityattimet=∞;see[Kr1],[KS]).Furthermore,Mitrea[Mit]usedaversionoftheVolterracalculusforstudyingparabolicequationswithDirichletboundaryconditionsonLipschitzdomainsandMikayelyan[Mi2]dealtwithparabolicproblemsonmanifoldswithedgesviaanextensionoftheVolterracalculustothesettingofSchulze’sedgecalculus([Sc1],[Sc2]).Ontheotherhand,in[Po2]theapproachtotheheatkernelasymptoticsofGreiner[Gr]wascombinedwiththerescalingofGetzler[Ge]toproduceanewshortproofofthelocalindexformulaofAtiyah-Singer[AS].TheupshotisthatthisproofisassimpleasGetzlershortproofin[Ge]but,unlikethelatter,itallowsustosimilarlycomputetheConnes-Moscovicicocycle[CM]forDiracspectraltriples.Furthermore,thepseudodifferentialrepresentationoftheheatkernelprovidedbytheVolterracalculusin[Gr]alsogivesanalternativetotheconstructionbySeeley[Se]ofpseudodifferentialcomplexpowersof(hypo)ellipticdifferentialoperators(cf.[Po1],[Po3];seealso[MSV]).WhilemostoftheusualpropertiesoftheclassicalΨDOcalculusholdverbatiminthesettingoftheVolterracalculus,amoredelicateissueistocheckasymptoticcompleteness.Thispropertyallowsustoconstructpara-metricesforparabolicoperators,butitsstandardproofcannotbecarriedthroughinthesettingoftheVolterracalculus.Indeed,atthelevelofsym-bolstheVolterrapropertycorrespondstoanalyticitywithrespecttothetimecovariable(seeSection1),butthispropertyisnotpreservedbythecut-offargumentsoftheproof.SincewecannotcutoffVolterrasymbols,Piriou[Pi1,pp.82–88]provedtheasymptoticcompletenessoftheVolterracalculusbycuttingoffdistributionkernelsinstead,whichatthisleveldoesnotharmtheVolterraproperty,andbycheckingthatundertheFouriertransformwegetanactualasymptoticexpansionofsymbols(seealso[Me]).Recently,Krainer[Kr2,pp.62–73]ob-tainedaproofbymakinguseofthekernelcut-offoperatorofSchulze([Sc1],[Sc2])andMikayelyan[Mi1]producedanotherproofbycombiningtransla-tionsinthetimecovariablewithaninductionprocess1.Inthispaper,wepresentsomewhatsimplerapproaches.First,weshowthatweactuallygetaVolterraΨDObyaddingasuitablesmoothingoperatortotheΨDOprovidedbythestandardproofoftheasymptoticcompletenessofclassicalsymbols(seeProposition2.1).1Despitethatin[Mi1,p.79]theinductionhypothesisisnotstatedproperlyandthereisatypoonline14theargumentintheproofiscorrect.2Second,wedealwiththeasymptoticcompletenessofanalyticVolterrasymbols(seeProposition3.3andProposition3.6).Thiswasthesettingunderconsiderationin[Kr2]and[Mi1],becausethisasymptoticcompletenessimpliesthatoftheVolterracalculus(seeSection3).HereourapproachisinspiredbytheversionoftheBorellemmaforanalyticfunctionsonanangularsector(e.g.[AG,p.63]).Thispaperisorganizedasfollows.InSection1webrieflyreviewthemainfactsconcerningtheVolterracalculus.InSection2wepresentourfirstap-proach.InSection3wecarryoutourproofsoftheasymptoticcompletenessofanalyticsymbols.Finally,intheappendix,writtenwithHaykMikayelyan,wegivealternativeproofsoftheasymptoticcompleteness