HardyMartingales ThisbookpresentstheprobabilisticmethodsaroundHardymartingalesforanaudience interestedinapplicationstocomplex,harmonic,andfunctionalanalysis.Buildingon theworkofBourgain,Garling,Jones,Maurey,Pisier,andVaropoulos,itdiscussesin detail those martingale spaces that reflect characteristic qualities of complex analytic functions. Its particular themes are holomorphic random variables on Wiener space, andHardymartingalesontheinfinitetorusproduct,andnumerousdeepapplicationsto ∞ the geometry and classification of complex Banach spaces, e.g., the SL estimates for Doob’s projection operator, the embedding of L1 into L1/H1, the isomorphic classificationtheoremforthepolydiskalgebras,ortherealvariablescharacterizationof BanachspaceswiththeanalyticRadonNikodymproperty.Duetotheinclusionofkey backgroundmaterialonstochasticanalysisandBanachspacetheory,it’ssuitablefora widespectrumofresearchersandgraduatestudentsworkinginclassicalandfunctional analysis. Paul F.X. Mu¨ller isProfessoratJohannesKeplerUniversityLinz,Austria.He istheauthorofmorethan50papersincomplex,harmonicandfunctionalanalysis,and ofthemonographIsomorphismsbetweenH1spaces(Springer,2005). Published online by Cambridge University Press NEW MATHEMATICAL MONOGRAPHS EditorialBoard JeanBertoin,Be´laBolloba´s,WilliamFulton,BrynaKra,IekeMoerdijk, CherylPraeger,PeterSarnak,BarrySimon,BurtTotaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversity Press.Foracompleteserieslistingvisitwww.cambridge.org/mathematics. 1. M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups 2. J.B.GarnettandD.E.MarshallHarmonicMeasure 3. P.CohnFreeIdealRingsandLocalizationinGeneralRings 4. E.BombieriandW.GublerHeightsinDiophantineGeometry 5. Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns 6. S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures 7. A.ShlapentokhHilbert’sTenthProblem 8. G.MichlerTheoryofFiniteSimpleGroupsI 9. A.BakerandG.Wu¨stholzLogarithmicFormsandDiophantineGeometry 10. P.KronheimerandT.MrowkaMonopolesandThree-Manifolds 11. B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T) 12. J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory 13. M.GrandisDirectedAlgebraicTopology 14. G.MichlerTheoryofFiniteSimpleGroupsII 15. R.SchertzComplexMultiplication 16. S.BlochLecturesonAlgebraicCycles(2ndEdition) 17. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups 18. T.DownarowiczEntropyinDynamicalSystems 19. C.SimpsonHomotopyTheoryofHigherCategories 20. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesI 21. E.FricainandJ.MashreghiTheTheoryofH(b)SpacesII 22. J.Goubault-LarrecqNon-HausdorffTopologyandDomainTheory 23. J.S´niatyckiDifferentialGeometryofSingularSpacesandReductionofSymmetry 24. E.RiehlCategoricalHomotopyTheory 25. B.A.MunsonandI.Volic´CubicalHomotopyTheory 26. B.Conrad,O.GabberandG.PrasadPseudo-reductiveGroups(2ndEdition) 27. J.Heinonen,P.Koskela,N.ShanmugalingamandJ.T.TysonSobolevSpacesonMetric MeasureSpaces 28. Y.-G.OhSymplecticTopologyandFloerHomologyI 29. Y.-G.OhSymplecticTopologyandFloerHomologyII 30. A.BobrowskiConvergenceofOne-ParameterOperatorSemigroups 31. K.CostelloandO.GwilliamFactorizationAlgebrasinQuantumFieldTheoryI 32. J.-H.EvertseandK.Gyo˝ryDiscriminantEquationsinDiophantineNumberTheory 33. G.FriedmanSingularIntersectionHomology 34. S.SchwedeGlobalHomotopyTheory 35. M.Dickmann,N.SchwartzandM.TresslSpectralSpaces 36. A.BaernsteinIISymmetrizationinAnalysis 37. A.Defant,D.Garc´ıa,M.MaestreandP.Sevilla-PerisDirichletSeriesandHolomorphic FunctionsinHighDimensions 38. N.Th.VaropoulosPotentialTheoryandGeometryonLieGroups 39. D.ArnalandB.CurreyRepresentationsofSolvableLieGroups 40. M.A.Hill,M.J.HopkinsandD.C.RavenelEquivariantStableHomotopyTheoryandthe KervaireInvariantProblem 41. K.CostelloandO.GwilliamFactorizationAlgebrasinQuantumFieldTheoryII 42. S.KumarConformalBlocks,GeneralizedThetaFunctionsandtheVerlindeFormula Published online by Cambridge University Press Hardy Martingales Stochastic Holomorphy, L1-Embeddings, and Isomorphic Invariants PAUL F.X. MU¨ LLER JohannesKeplerUniversityLinz Published online by Cambridge University Press UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108838672 DOI:10.1017/9781108976015 ©PaulF.X.Mu¨ller2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-83867-2Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Published online by Cambridge University Press ToJoanna Published online by Cambridge University Press Published online by Cambridge University Press Contents Preface pageix Acknowledgments xv 1 StochasticHolomorphy 1 1.1 Preliminaries 1 1.2 HolomorphicMartingales 35 1.3 Extrapolation 50 1.4 StochasticHilbertTransforms 54 1.5 MartingaleEmbeddingandProjection 58 1.6 ProjectingHolomorphicMartingales 60 1.7 ApplicationstoHp(T) 66 1.8 ProjectingSquareFunctions 68 1.9 Notes 84 2 HardyMartingales 87 2.1 Bochner–LebesqueSpaces 87 2.2 MartingalesonTN 94 2.3 Examples 100 2.4 ClassesofMartingalesandProjections 109 2.5 BasicL1Estimates 125 2.6 Notes 134 3 EmbeddingL1inL1/H1 136 0 3.1 HardyMartingalesandDyadicPerturbations 136 3.2 TheQuotientSpaceL1(T)/H1(T) 156 0 3.3 Notes 171 4 EmbeddingL1inXorL1/X 172 4.1 RandomMeasuresRepresentingOperators 172 4.2 L1Embedding 205 4.3 Rosenthal’sL1-Theorem 225 vii Published online by Cambridge University Press viii Contents 4.4 ApproximationinL1 241 4.5 Talagrand’sExamples 258 4.6 Notes 272 5 IsomorphicInvariants 274 5.1 IsomorphicClassificationofPolydiskAlgebras 275 5.2 HardyMartingaleConvergenceaRNP 304 5.3 HardyMartingaleCotype 340 5.4 Unconditionality(aUMD) 356 5.5 Embedding 371 5.6 Interpolation 375 5.7 Notes 378 Appendix:RNP,UMD,andM-Cotype 381 6 OperatorsonLp(cid:0)L1(cid:1) 391 6.1 TheHilbertTransform 392 6.2 ReflexiveSubspacesofL1 407 6.3 Notes 425 7 FormativeExamples 427 7.1 L1QuotientsbyReflexiveSpaces 428 7.2 TheTraceClass 440 7.3 IteratedLp(Lq)Spaces 451 7.4 TheJamesTree 458 7.5 TheSpaceL1/H1 474 0 7.6 Notes 481 References 483 NotationIndex 497 SubjectIndex 498 Published online by Cambridge University Press Preface In this book we present probabilistic methods developed for applications to complexandfunctionalanalysis.Wewillstudy,indepth,spacesofmartingales thatreflectcharacteristicqualitiesofholomorphicfunctions;specifically, (i) thespaceofintegrableHardymartingales H1(cid:0)TN(cid:1)definedbyrestrictions onthesupportoftheirFouriercoefficients; (ii) the space of holomorphic random variables H1(Ω) on Wiener space, characterizedbytheirItoˆ-integralrepresentation. StochasticHolomorphy The interplay between (nonconstant) holomorphic functions f and complex Brownian motion (z) goes back to the work of Paul Le´vy who noted that t f(z) is the path of a complex Brownian motion (Le´vy, 1948). More pre- t cisely, f(z)isdistributionallyindistinguishablefromz where t β(t) (cid:90) t β(t)= f(cid:48)(zs)2ds. | | 0 Itoˆ and McKean (1965) presented a proof of Picard’s theorem (asserting that a nonconstant analytic function on C omits at most one value) by applying Le´vy’sresulttotheuniversalcoveringmapontoC 0,1 . \{ } Beingmorespecific,welet f: D CbeanalyticandboundedwhereD = → z C : z < 1 . We let τ denote the exit time of (z) from D. The process t { ∈ | | } (f(z):t<τ)maybeexpandedbyItoˆ integrals, t (cid:90) t f(zt)= f(cid:48)(zs)dzs, t<τ, (0.0.1) 0 and hence forms a Brownian martingale. Doob (1953) proved martingale convergence theorems, showing that lim f(z) exists almost surely, and t τ t → ix https://doi.org/10.1017/9781108976015.001 Published online by Cambridge University Press x Preface developed the tools (conditioned Brownian motion) by which martingale convergenceistransformedintoradiallimitssuchthat lim f(rζ) existsforalmostevery ζ T, r 1 ∈ → whereT= z C: z =1 .ThusFatou’stheoremandPrivalov’stheoremwere { ∈ | | } among the first results in complex analysis obtained by stochastic methods. ManyyearslaterBurkholderetal.(1971)showedthat E f(z ) CEsup f(z ), τ s | |≤ |(cid:60) | s<τ whichinturn,byDoob’sconditionedBrownianmotion,givesrisetothereal variablecharacterizationoftheHardyspaceH1(T). TheItoˆintegralrepresentation(0.0.1)providesthemodelforanintrinsically stochastic concept of holomorphy. We say that F : Ω C is a holomorphic t → martingaleonWienerspace(Ω,( ),P)ifthereexistsacomplex-valued,( ) t t F F adaptedprocess(X)satisfying t (cid:90) t F = F + X dz . t 0 s s 0 AholomorphicrandomvariableisjustthelimitF =lim F ofanequiinte- t t →∞ grableholomorphicmartingale. In applications to analysis, it is important that holomorphic martingales form a linear space and that they are stable under stopping times, pointwise multiplication, and composition with analytic functions. Varopoulos (1980, 1981) found a key to profound probabilistic methods in complex analysis by analyzingtheactionofDoob’sconditionalexpectationoperator, N(F)(cid:0)eiθ(cid:1)=E(cid:16)Fz =eiθ(cid:17), τ | on a holomorphic random variable F. It consists in linking the space of p-integrableholomorphicrandomvariablesHp(Ω)tothecorrespondingHardy spaceHp(T)bymeansofthecommutativediagram, Hp(T) Id (cid:47)(cid:47) H(cid:58)(cid:58) p(T) , M (cid:36)(cid:36) N Hp(Ω) whereMf = f(z ).Chapter1developscentralideasofstochasticholomorphy τ together with their applications to the Marcinkiewicz decomposition and to complex convexity estimates for Hp(T) and for the Banach space SL (T) of ∞ harmonicfunctionswithuniformlyboundedsquarefunctions. https://doi.org/10.1017/9781108976015.001 Published online by Cambridge University Press