Hardy Martingales: Stochastic Holomorphy, L^1-Embeddings, and Isomorphic Invariants

516 Pages·2022·9.259 MB·English

by  Paul F. X. Müller

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).
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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
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Informationonthistitle:www.cambridge.org/9781108838672
DOI:10.1017/9781108976015
©PaulF.X.Mu¨ller2022
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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
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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
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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