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Lecture Notes in Mathematics 2326 Steven G. Krantz The E. M. Stein Lectures on Hardy Spaces Lecture Notes in Mathematics Volume 2326 Editor-in-Chief Jean-MichelMorel,CMLA,ENS,Cachan,France BernardTeissier,IMJ-PRG,Paris,France SeriesEditors KarinBaur,UniversityofLeeds,Leeds,UK MichelBrion,UGA,Grenoble,France AnnetteHuber,AlbertLudwigUniversity,Freiburg,Germany DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK AngelaKunoth,UniversityofCologne,Cologne,Germany ArianeMézard,IMJ-PRG,Paris,France MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg MarkPolicott,MathematicsInstitute,UniversityofWarwick,Coventry,UK SylviaSerfaty,NYUCourant,NewYork,NY,USA László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany GabrieleVezzosi,UniFI,Florence,Italy AnnaWienhard,RuprechtKarlUniversity,Heidelberg,Germany This series reports on new developments in all areas of mathematics and their applications-quickly,informallyandatahighlevel.Mathematicaltextsanalysing newdevelopmentsinmodellingandnumericalsimulationarewelcome.Thetypeof materialconsideredforpublicationincludes: 1. Researchmonographs 2. Lecturesonanewfieldorpresentationsofanewangleinaclassicalfield 3. Summerschoolsandintensivecoursesontopicsofcurrentresearch. Textswhichareoutofprintbutstillindemandmayalsobeconsiderediftheyfall withinthesecategories.Thetimelinessofamanuscriptissometimesmoreimportant thanitsform,whichmaybepreliminaryortentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews,andzbMATH. Steven G. Krantz The E. M. Stein Lectures on Hardy Spaces StevenG.Krantz DepartmentofMathematics WashingtonUniversity St.Louis,MO,USA ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-031-21951-1 ISBN978-3-031-21952-8 (eBook) https://doi.org/10.1007/978-3-031-21952-8 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2023 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 TothememoryofE.M.Stein,forhis teachingandhisfriendship. Preface Elias M. Stein was one of the pre-eminent harmonic analysts of the twentieth century.Hedirectedmorethan50Ph.D.studentsatPrincetonUniversity,andmany ofthemarequitedistinguishedmathematiciansintheirownright.Sohisinfluence continuesonintothetwenty-firstcentury. OneofStein’sseminalcontributionstomodernmathematicalanalysisisthereal variable theory of Hardy spaces. Hardy spaces were first developed by F. Riesz and M. Riesz in the early part of the twentieth century. It was in fact F. Riesz who, in 1923, named these spaces in honor of G. H. Hardy. These were spaces ofholomorphicfunctionsontheunitdiscinthecomplexplane.TheHardyspaces, orHp,areimportantbecauseoftheirstructuralpropertiesandalsobecauseoftheir behaviorundercertainimportantmappings. People long suspected that, lurking in the background, there is a real-variable theory of Hardy spaces—one that rejects the complex variable context in which theywereoriginallyformulated.Arealvariabletheorywouldallowustofocuson theessentialstructureandmappingpropertiesofthesespaces. The pioneering work along these lines was done by E. M. Stein and Guido L. Weiss in 1960. There they realized that the right way to formulate a real-variable Hp space was as the gradients of harmonic functions. A number of important calculationsandinsightsappearintheirActaMathematicapaper. A1971paperbyBurkholderetal. [BGS]providesaglimpseofwhatispossible inthisdirection.Thispaperdependsdecisivelyonprobabilisticmethods. Thepaperthatbrokethesubjectwideopenwasthe1972paperofC.Fefferman andE.M.Stein.Thispaperisoneofthemosthighlycitedinmodernmathematics.It inspiredafloodofworkinthe1970sandonintothe1980s,andcontinuestohavea stronginfluencetoday.AmajorAMSSummerWorkshopwasheldinWilliamstown, Massachusettsin1978tocelebrateandstudythispaperofFefferman/Steinandits consequences. The present book is based on a year-long course that Stein taught at Princeton University in 1973–1974. He in fact taught the course at the request of Robert Fefferman and this author. It was a huge and protracted effort for him to produce thiscourseonthespot,andtheresultswerestunning. vii viii Preface The course that Stein taught was wide-ranging, deep, and insightful. In the characteristic Stein manner, he not only stated theorems and proved them but also worked examples, formulated conjectures, and handed out open problems. Every lecturewasanobjectlessonandavaluablecommodity. ThisauthorwroteupverycarefulnotesoftheSteincourseonHardyspaces.In fact he wrote up notes while he was sitting in the course in Fine Hall. But then, a year or two later, he rewrote and polished the notes a second time. The book that we are presenting here is a formal development of those notes. The purpose now is to share with the mathematical world the perspective of Stein on this subject areathatheinvented.Itisaglowinglookbackatoneofthemilestonesofmodern mathematics.ItisatributetoE.M.Stein. The content of E. M. Stein’s 1973–1974 lecture course, “Real Variable Hardy Spaces,”hasbeenreproducedbyStevenG.KrantzwithpermissionfromtheStein estate. Ofcourseallerrorsandmis-stepscontainedhereinaretheresponsibilityofthe author.Welookforwardtohearingfromreadersasthebookisreadandappreciated. St.Louis,Mo,USA StevenG.Krantz Contents 1 IntroductoryMaterial ...................................................... 1 1.1 VariousMaximalFunctions ............................................ 1 1.2 NontangentialConvergence ............................................ 6 1.3 UnrestrictedConvergence .............................................. 16 1.4 TheAreaIntegral ....................................................... 19 1.5 GeneralizationsofR+andHp(R+)................................... 22 1.6 RelationshipsAmongDomains ........................................ 33 2 MoreonHardySpaces ...................................................... 35 2.1 HardySpacesandMaximalFunctions................................. 35 2.2 MoreMaximalFunctions............................................... 43 2.3 RealVariableHp ....................................................... 51 2.4 SomeThoughtsonSummability ....................................... 59 3 BackgroundonHp Spaces ................................................. 73 3.1 WhereDidHp SpacesGetStarted? ................................... 73 3.2 HardySpacesinC1 ..................................................... 76 3.3 TheHardy–LittlewoodMaximalFunction............................. 86 3.4 ThePoissonKernelandFourierInversion............................. 93 4 HardySpacesonD .......................................................... 101 4.1 TheRoleoftheHilbertTransform..................................... 101 4.2 BlaschkeProducts....................................................... 120 4.3 PassagefromDtoR2+.................................................. 129 5 HardySpacesonRn ......................................................... 137 5.1 ThePoissonKernelontheBall ........................................ 137 5.2 ThePoissonKernelontheUpperHalfspace .......................... 140 5.3 Cauchy–RiemannSystems ............................................. 149 5.4 ACharacterizationofHp,p >1...................................... 159 5.5 TheAreaIntegral ....................................................... 163 5.6 ApplicationsoftheMaximalFunctionCharacterization.............. 179 5.7 H1(Rn)andDualitywithRespecttoBMO............................ 206 ix x Contents 6 DevelopmentsSince1974.................................................... 227 6.1 TheAtomicTheory..................................................... 227 6.2 TheLocalTheoryofHardySpaces.................................... 228 6.3 The Work of Chang/Krantz/Stein on HardySpaces for EllipticBoundaryValueProblems ..................................... 231 6.4 Multi-ParameterHarmonicAnalysis................................... 232 6.5 TheT1TheoremofDavid/Journé ..................................... 233 6.6 ContributionsofTomWolff ............................................ 233 6.7 Wavelets................................................................. 234 6.7.1 LocalizationintheTimeandSpaceVariables................. 234 6.7.2 BuildingaCustomFourierAnalysis........................... 235 6.7.3 TheHaarBasis.................................................. 237 7 ConcludingRemarks ........................................................ 243 References......................................................................... 245 Index............................................................................... 249