Spaces of Holomorphic Functions in the Unit Ball

280 Pages·2005·2.571 MB·English

by  Kehe Zhu

226
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
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(continued after index)
Kehe Zhu
Spaces of Holomorphic
Functions in the Unit Ball
Springe]
Kehe Zhu
Department of Mathematics
State University of New York at Albany
Albany, NY 12222
USA
kzhu @ math.albany.edu
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East Hall University of California,
University University of Michigan Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840
USA USA USA
[e-mail] fgehring @ math.lsa.umich.edu ribet @ math .berkeley. edu
Mathematics Subject Classification (2000): MSCM12198, SCM12198, SCM12007
Library of Congress Cataloging-in-Publication Data
Zhu, Kehe, 1961-
Spaces of holomorphic functions in the unit ball / Kehe Zhu.
p. cm.
Includes bibliographical references and index.
ISBN 0-387-22036-4 (alk. paper)
1. Holomorphic functions. 2. Unit ball. I. Title.
QA331.Z48 2004
515^98—dc22 2004049191
ISBN 0-387-22036-4 Printed on acid-free paper.
© 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New
York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
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The use in this publication of trade names, trademarks, service marks, and similar terms, even if
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Printed in the United States of America. (EB)
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To my family: Peijia, Peter, and Michael
Preface
Therehasbeenaflurryofactivityinrecentyearsinthelooselydefinedareaofholo-
morphicspaces.Thisbookdiscussesthemostwell-knownandwidelyusedspacesof
holomorphicfunctionsintheunitballofCn:Bergmanspaces,Hardyspaces,Besov
spaces,Lipschitzspaces,BMOA,andtheBlochspace.
Thethemeofthebookisverysimple.Foreachscaleofspaces,Idiscussintegral
representations,characterizationsintermsofvariousderivatives,atomicdecomposi-
tion,complexinterpolation,andduality.Veryfewotherpropertiesarediscussed.
I chose the unit ball as the setting because most results can be achieved there
using straightforwardformulaswithoutmuchfuss. In fact, most of the results pre-
sentedinthebookarebasedontheexplicitformoftheBergmanandCauchy-Sze¨go
kernels.Thebookcanbereadcomfortablybyanyonefamiliarwithsinglevariable
complexanalysis;noprerequisiteonseveralcomplexvariablesisrequired.
Few of the results in the book are new, but most of the proofs are originally
constructedandconsiderablysimplerthantheexistingonesin theliterature.There
is some obvious overlap between this book and Walter Rudin’s classic “Function
TheoryintheUnitBallofCn”.Buttheoverlapisnotsubstantial,anditismyhope
thatthetwobookswillcomplementeachother.
The book is essentially self-contained, with two exceptions worth mentioning.
First,theexistenceofboundaryvaluesforfunctionsintheHardyspacesHpisproved
only for p ≥ 1; a full proof can be found in Rudin’s book. Second, the complex
interpolationbetweentheHardyspacesH1andHp(orBMOA)isnotproved;afull
proofrequiresmorerealvariabletechniques.
The exercises at the end of each chapter vary greatly in the level of difficulty.
Someofthemaresimpleapplicationsofthemaintheorems,someareobviousgen-
eralizationsorvariations,whileothersaredifficultresultsthatcomplementthemain
text.Inthelattercase,atleastonereferenceisprovidedforthereader.
I apologize in advance for any misrepresentation in the short sections entitled
“Notes”,foranyomissionofsignificantreferences,andforhavingnotincludedone
orseveralofyourfavoritetheorems.Ididnoteventrytocompileacomprehensive
bibliography.
VIII Preface
Thetopicschosenforthebook,andthewaytheyareorganized,reflectentirely
my own taste, preference/prejudice,and research/teaching experience. Among the
topics that I thought about seriously but eventually decided not to include are the
so-calledArvesonspace,theso-calledQ spaces,andgeneralholomorphicSobolev
p
spaces. Of course, the Bergman spaces, the Bloch space, the holomorphic Besov
spaces, and the holomorphic Lipschitz spaces can all be considered special cases
of a moregeneralfamilyofholomorphicSobolevspaces.Itappearsto me thatthe
relativelyeleganttreatmentofthesespecialcasesismoreinterestingandappealing
thananotherwisemorecumbersomepresentationofanexhaustiveclassoffunctions.
During the preparationof the manuscriptI received help and advice from Boo
Rim Choe, Joe Cima, Richard Rochberg,and Jie Xiao. It is my pleasure to record
mythankstothemhere.IamparticularlygratefultoRuhanZhao,whoreadaprelim-
inaryversionoftheentiremanuscriptandcaughtnumerousmisprintsandmistakes.
Myfamily—Peijia,Peter,andMichael—providedmewithlove,understanding,and
blocksofuninterruptedtimethatisnecessaryforthecompletionofanymathematical
project.
Albany,June2004 KeheZhu
Contents
1 Preliminaries ................................................ 1
1.1 HolomorphicFunctions ..................................... 1
1.2 TheAutomorphismGroup................................... 3
1.3 LebesgueSpaces ........................................... 9
1.4 SeveralNotionsofDifferentiation............................. 17
1.5 TheBergmanMetric........................................ 22
1.6 TheInvariantGreen’sFormula ............................... 28
1.7 SubharmonicFunctions ..................................... 31
1.8 InterpolationofBanachSpaces ............................... 33
Notes ......................................................... 35
Exercises ...................................................... 35
2 BergmanSpaces ............................................. 39
2.1 BergmanSpaces ........................................... 39
2.2 BergmanTypeProjections ................................... 43
2.3 OtherCharacterizations ..................................... 48
2.4 CarlesonTypeMeasures..................................... 56
2.5 AtomicDecomposition...................................... 62
2.6 ComplexInterpolation ...................................... 73
Notes ......................................................... 74
Exercises ...................................................... 75
3 TheBlochSpace ............................................. 79
3.1 TheBlochspace ........................................... 79
3.2 TheLittleBlochSpace ...................................... 89
3.3 Duality ................................................... 93
3.4 Maximality................................................ 94
3.5 PointwiseMultipliers ....................................... 97
3.6 AtomicDecomposition...................................... 98
3.7 ComplexInterpolation ...................................... 103
Notes ......................................................... 104
X Contents
Exercises ...................................................... 104
4 HardySpaces................................................109
4.1 ThePoissonTransform...................................... 109
4.2 HardySpaces.............................................. 122
4.3 TheCauchy-Szego¨Projection ................................ 131
4.4 SeveralEmbeddingTheorems................................ 141
4.5 Duality ................................................... 149
Notes ......................................................... 152
Exercises ...................................................... 152
5 FunctionsofBoundedMeanOscillation..........................157
5.1 BMOA ................................................... 157
5.2 CarlesonMeasures ......................................... 162
5.3 VanishingCarlesonMeasuresandVMOA...................... 169
5.4 Duality ................................................... 172
5.5 BMOintheBergmanMetric ................................. 180
5.6 AtomicDecomposition...................................... 185
Notes ......................................................... 194
Exercises ...................................................... 194
6 BesovSpaces ................................................199
6.1 TheSpacesB ............................................. 199
p
6.2 TheMinimalMo¨biusInvariantSpace.......................... 204
6.3 Mo¨biusInvarianceofB .................................... 207
p
6.4 TheDirichletSpaceB2 ..................................... 210
6.5 DualityofBesovSpaces..................................... 216
6.6 OtherCharacterizations ..................................... 225
Notes ......................................................... 230
Exercises ...................................................... 230
7 LipschitzSpaces .............................................235
7.1 B Spaces ................................................ 235
α
7.2 TheLipschitzSpacesΛ for0<α<1........................ 241
α
7.3 TheZygmundClass ........................................ 246
7.4 Thecaseα>1 ............................................ 247
7.5 AUnifiedTreatment........................................ 250
7.6 GrowthinTangentialDirections .............................. 253
7.7 Duality ................................................... 257
Notes ......................................................... 259
Exercises ...................................................... 260
References ......................................................263
Index...........................................................269
1
Preliminaries
Inthischapterwesetthestageanddiscussthebasicpropertiesoftheunitball.Sev-
eralresultsandtechniquesofthischapterwillbeusedrepeatedlyin laterchapters.
Theseincludethechangeofvariablesformula,thefractionaldifferentialandintegral
operators,thebasicintegralestimateofthekernelfunctions(Theorem1.12),andthe
Marcinkiewiczinterpolationtheorem.Also,theradialderivative,theinvariantLapla-
cian,theautomorphismgroup,andtheBergmanmetricareessentialconceptsforthe
restofthebook.
1.1 HolomorphicFunctions
Let C denote the set of complex numbers. Throughoutthe book we fix a positive
integernandlet
Cn =C×···×C
denotetheEuclideanspaceofcomplexdimensionn.Addition,scalarmultiplication,
andconjugationaredefinedonCncomponentwise.For
z =(z1,···,zn), w =(w1,···,wn),
inCn,wedefine
(cid:2)z,w(cid:3)=z1w1+···+znwn,
wherew isthecomplexconjugateofw .Wealsowrite
k k
(cid:1) (cid:1)
|z|= (cid:2)z,z(cid:3) = |z1|2+···+|zn|2.
ThespaceCn becomesann-dimensionalHilbertspacewhenendowedwiththe
innerproductabove.ThestandardbasisforCnconsistsofthefollowingvectors:
e1 =(1,0,0,···,0), e2 =(0,1,0,···,0), ···, en =(0,0,···,0,1).