Geometric Harmonic Analysis I: A Sharp Divergence Theorem with Nontangential Pointwise Traces PDF
Preview Geometric Harmonic Analysis I: A Sharp Divergence Theorem with Nontangential Pointwise Traces
Developments in Mathematics Dorina Mitrea Irina Mitrea Marius Mitrea Geometric Harmonic Analysis I A Sharp Divergence Theorem with Nontangential Pointwise Traces Developments in Mathematics Volume 72 SeriesEditors KrishnaswamiAlladi,DepartmentofMathematics,UniversityofFlorida, Gainesville,FL,USA PhamHuuTiep,DepartmentofMathematics,RutgersUniversity,Piscataway,NJ, USA LoringW.Tu,DepartmentofMathematics,TuftsUniversity,Medford,MA,USA AimsandScope TheDevelopmentsinMathematics(DEVM)bookseriesisdevotedtopublishing well-written monographs within the broad spectrum of pure and applied mathe- matics. Ideally, each book should be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential rela- tionship to other fields are most welcome. High-quality edited volumes conveying currentstate-of-the-artresearchwilloccasionallyalsobeconsideredforpublication. TheDEVMseriesappealstoavarietyofaudiencesincludingresearchers,postdocs, andadvancedgraduatestudents. · · Dorina Mitrea Irina Mitrea Marius Mitrea Geometric Harmonic Analysis I A Sharp Divergence Theorem with Nontangential Pointwise Traces DorinaMitrea IrinaMitrea DepartmentofMathematics DepartmentofMathematics BaylorUniversity TempleUniversity Waco,TX,USA Philadelphia,PA,USA MariusMitrea DepartmentofMathematics BaylorUniversity Waco,TX,USA ISSN 1389-2177 ISSN 2197-795X (electronic) DevelopmentsinMathematics ISBN 978-3-031-05949-0 ISBN 978-3-031-05950-6 (eBook) https://doi.org/10.1007/978-3-031-05950-6 MathematicsSubjectClassification: 26A16, 26A46, 26B20, 28A25, 28A75, 28A78, 28C15, 30G35, 31B10,31C12,42B25,42B37,49Q15,53B20,53C65,58A10,58C35 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Dedicatedwithlovetoourparents Prefacing the Full Series The current work is part of a series, comprised of five volumes. In broad terms, theprincipalaimistodeveloptoolsinRealandHarmonicAnalysis,ofgeometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problemsformulatedinrathergeneralgeometricandanalyticsettings. InVolumeIweestablishasharpversionofDivergenceTheorem(akaFundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundarytraceisonlyassumedtoexistinanontangentialpointwisesense. Volume II is concerned with function spaces measuring size and/or smooth- ness,suchasHardyspaces,Besovspaces,Triebel–Lizorkinspaces,Sobolevspaces, Morrey spaces, Morrey–Campanato spaces, and spaces of functions of Bounded Mean Oscillations, in general geometric settings. Work here also highlights the closeinterplaybetweendifferentiabilitypropertiesoffunctionsandsingularintegral operators. ThetopicofsingularintegraloperatorsisproperlyconsideredinVolumeIII,where wedevelopaversatileCalderón–Zygmundtheoryforsingularintegraloperatorsof convolutiontype(andwithvariablecoefficientkernels)onuniformlyrectifiablesets intheEuclideanambient,andthesettingofRiemannianmanifolds.Applicationsto scatteringbyroughobstaclesarealsodiscussedinthisvolume. InVolumeIVwefocusonsingularintegraloperatorsofboundarylayertypewhich enjoymorespecializedproperties(comparedwithgeneric,gardenvarietysingular integraloperatorstreatedearlierinVolumeIII).ApplicationstoComplexAnalysisin severalvariablesaresubsequentlypresented,startingfromtherealizationsthatmany natural integral operators in this setting, such as the Bochner–Martinelli operator, are actual particular cases of double layer potential operators associated with the complexLaplacian. InVolumeV,whereeverythingcomestogether,finerestimatesforacertainclass ofsingularintegraloperators(ofchord-dot-normaltype)areproducedinamanner whichindicateshowtheirsizeisaffectedbythe(infinitesimalandglobal)flatness of the “surfaces” on which they are defined. Among the library of double layer potential operators associated with a given second-order system, we then identify thosedoublelayerswhichfallunderthiscategoryofsingularintegraloperators.It vii viii PrefacingtheFullSeries is precisely for this subclass of double layer potentials that Fredholm theory may then be implemented assuming the underlying domain has a compact boundary, whichissufficientlyflatatinfinitesimalscales.Fordomainswithunboundedbound- aries,thisvery category ofdouble layer potentials may beoutrightinverted,using a Neumann series argument, assuming the “surface” in question is sufficiently flat globally. In turn, this opens the door for solving a large variety of boundary value problems for second-order systems (involving boundary data from Muckenhoupt weighted Lebesgue spaces, Lorentz spaces, Hardy spaces, Sobolev spaces, BMO, VMO, Morrey spaces, Hölder spaces, etc.) in a large class of domains which, for example,areallowedtohavespiralsingularities(hencemoregeneralthandomains locallydescribedasupper-graphsoffunctions).Intheoppositedirection,weshow thattheboundaryvalueproblemsformulatedforsystemslackingsuchspeciallayer potentialsmayfailtobeFredholmsolvableevenforreallytamedomains,likethe upper half-space, or the unit disk. Save for the announcement [184], all principal resultsappearhereinprintforthefirsttime. Weclosewithashortepilogue,attemptingtoplacetheworkundertakeninthis series into a broader picture. The main goal is to develop machinery of geometric harmonicanalysisflavorcapableofultimatelydealingwithboundaryvalueproblems ofaverygeneralnature.Oneoftheprincipaltools(indeed,thepiecèderésistance) inthisregardisanewandpowerfulversionoftheDivergenceTheorem,devisedin Volume I, whose very formulation has been motivated and shaped from the outset by its eventual applications to Harmonic Analysis, Partial Differential Equations, PotentialTheory,andComplexAnalysis.Thefactthatitsfootprintsmaybeclearly recognizedinthemakeupofsuchadiversebodyofresults,aspresentedinVolumes II–V,servesasatestamenttotheversatilityandpotencyofourbrandofDivergence Theorem.Alas,ourenterpriseismultifaceted,soitssuccessiscruciallydependent onmanyotherfactors.Foronething,itisnecessarytodeveloparobustCalderón– Zygmundtheoryforsingularintegralsofboundarylayertype(aswedoinVolumes III–IV),associatedwithgenericweaklyellipticsystems,capableofaccommodating alargevarietyoffunctionspacesofinterestconsideredinratherinclusivegeometric settings(ofthesortdiscussedinVolumeII).Thisrendersthese(boundary-to-domain) layerpotentialsusefulmechanismsforgeneratinglotsofnull-solutionsforthegiven systemofpartialdifferentialoperators,whoseformatiscompatiblewiththedemands in the very formulation of the boundary value problem we seek to solve. Next, in ordertobeabletosolvetheboundaryintegralequationtowhichmattersarereduced in this fashion, the success of employing Fredholm theory hinges on the ability to suitablyestimatetheessentialnormsofthe(boundary-to-boundary)layerpotentials. In this vein, we succeed in relating the distance from such layer potentials to the spaceofcompactoperatorstotheflatnessoftheboundaryofthedomaininquestion (measuredintermsofinfinitesimalmeanoscillationsoftheunitnormal)inadesirable mannerwhichshowsthat,inaprecisequantitativefashion,theflatterthedomain,the smallertheproximitytocompactoperators.Thissubtleandpowerfulresult,bridging betweenanalysisandgeometry,mayberegardedasafar-reachingextensionofthe pioneeringworkofRadonandCarlemanintheearly1900s. PrefacingtheFullSeries ix Ultimately, our work aligns itself with the program stemming from A. P. Calderón’s1978ICMplenaryaddressinwhichheadvocatestheuseoflayerpoten- tials“formuchmoregeneralellipticsystems[thantheLaplacian]”,see[36,p.90], andmayberegardedasanoptimalextensionofthepioneeringworkofE.B.Fabes, M.Jodeit,andN.M.Rivièrein[81](wherelayerpotentialmethodshavebeenfirst usedtosolveboundaryvalueproblemsfortheLaplacianinboundedC1domains).In thisendeavor,wehavebeenalsomotivatedbytheproblem1posedbyA.P.Calderón on[36,p.95],askingtoidentifythefunctionspacesonwhichsingularintegraloper- ators(ofboundarylayertype)arewell-definedandcontinuous.Thisisrelevantsince, asCalderónmentions,“Aclarificationofthisquestionwouldbeveryimportantin thestudyofboundaryvalueproblemsforellipticequations[inroughdomains].The methodsemployedsofarseemtobeinsufficientforthetreatmentoftheseproblems.” We also wish to mention that our work is also in line with the issue raised as an open problem by C. Kenig in [147, Problem 3.2.2, pp. 116–117], where he asked whetheroperatorsoflayerpotentialtypemaybeinvertedonappropriateLebesgue and Sobolev spaces in suitable subclasses on NTA domains with compact Ahlfors regularboundaries. Thetaskofmakinggeometryandanalysisworkinunisonisfraughtwithdiffi- culties, and only seldom can a two-way street be built on which to move between these two worlds without loss of information. Given this, it is actually surprising thatinmanyinstanceswecomeveryclosetohavingoptimalhypotheses,almostan accurateembodimentofthesloganifitmakessensetowriteit,thenit’strue. Waco,TX,USA DorinaMitrea Philadelphia,PA,USA IrinaMitrea Waco,TX,USA MariusMitrea March2022 1Inthelastsectionof[36],simplytitled“Problems,”Calderónsinglestwodirectionsforfurther study.Thefirstoneist(cid:2)hefamousquestion(cid:3)whetherthesmallnessconditionon(cid:2)a(cid:3)(cid:2)L∞(theLipschitz constantofthecurve (x,a(x)): x ∈R onwhichheprovedtheL2-boundednessoftheCauchy operator)mayberemoved(asiswellknown,thishasbeensolvedintheaffirmativebyCoifman, McIntosh,andMeyerin[53]).Wearereferringheretothesecond(andfinal)problemformulated byCalderónon[36,p.95]. Acknowledgements The authors gratefully acknowledge partial support from the Simons Foundation (throughgrants#426669,#318658,#616050,#637481),aswellasNSF(grant# 1900938).PortionsofthisworkhavebeencompletedatBaylorUniversityinWaco, Temple University in Philadelphia, the Institute for Advanced Study in Princeton, MSRIinBerkeley,andtheAmericanInstituteofMathematicsinSanJose.Wewish to thank these institutions for their generous hospitality. Last but not the least, we aregratefultoMichaelE.Taylorforgentlyyetpersistentlyencouragingusoverthe yearstocompletethisproject. xi Description of Volume I Whatsortofanalysiscanacertaingeometricenvironmentsupport2?Whatkindof geometryisrequiredtoensuretheveracityofaspecificanalyticalresult3?Thisseries, comprisedoffivevolumes,isajourneyintoGeometricHarmonicAnalysis,abrand ofharmonicanalysis4ofdefinitegeometricflavor,whoseultimategoalistobuildthe necessarymachinerycapableofdealingwithproblemsinvolvingPartialDifferential Equationsinverygeneralsettings.Thelinchpinofthisenterpriseisanew,powerful andadaptable,higher-dimensionalversionoftheFundamentalTheoremofCalculus. If we were to summarize the key message of Volume I in just a few words, it wouldsimplyread: it’stimetoredefinewhattheDivergenceTheoremcando! Itseemsfaintlymiraculousthatafterthreecenturiesofbeinginthelimelightthis remainsanactiveareaofresearch,buttherealityofthematteristhatthisisademand- drivensubject.Indeed,progressinacertainfieldoftenrequiresyetanew,andever morepotent,brandofDivergenceTheorem,whichcanaccommodatecertainspecific features. For example, to deal with Plateau’s problem, R. Caccioppoli and E. De Giorgi haveintroducedtheclassofsetsoflocallyfiniteperimeter,anenvironmentinwhich E.DeGiorgiandH.Federerhavesubsequentlyproducedamagnificentversionof the Divergence Theorem. Alas, the class of vector fields to which the De Giorgi– FedererDivergenceTheoremapplies,smoothandcompactlysupportedintheentire Euclideanspace(inparticular,completelyunrelatedtotheoriginaldomain),isfartoo 2IsthereaHardy–Littlewoodmaximalinequality,aPoincaréinequality,aFundamentalTheorem of Calculus, a rich function space theory, etc.? Also, in a given setting, how can one measure smoothness of functions, what sort of operators are natural to consider, what type of boundary valueproblemsarewell-posedorFredholmsolvable,etc.? 3For example, one may seek geometrical conditions guaranteeing that certain singular integral operators(ofboundarylayertype)arebounded,orFredholm,orinvertible,onavarietyoffunction spacesofinterest. 4Classicallyunderstoodasthebreakingupofawholeintoitspartsastoelucidatetheirnature. xiii
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