Spaces of Holomorphic Functions in the Unit Ball PDF
Preview Spaces of Holomorphic Functions in the Unit Ball
226 Graduate Texts in Mathematics Editorial Board S. Axler F.W. Gehring K.A. Ribet Graduate Texts in Mathematics 1 TAKEUTI/ZARING. Introduction to 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nd ed. 2 OxTOBY. Measure and Category. 2nd ed. 35 ALEXANDER/WERMER. Several Complex 3 ScHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed. 2nd ed. 36 KELLEY/NAMIOKA et al. Linear 4 HILTON/STAMMBACH. A Course in Topological Spaces. Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAC LANE. Categories for the Working 38 GRAUERT/FRITZSCHE. Several Complex Mathematician. 2nd ed. Variables. 6 HUGHES/PIPER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras. 7 J.-P. SERRE. A Course in Arithmetic. 40 KEMENY/SNELL/KNAPP. Denumerable 8 TAKEUTI/ZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. Modular Functions and Dirichlet and Representation Theory. Series in Number Theory. 10 COHEN. A Course in Simple Homotopy 2nd ed. Theory. 42 J.-P. SERRE. Linear Representations of 11 CONWAY. Functions of One Complex Finite Groups. Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions. 13 ANDERSON/FULLER. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry. of Modules. 2nd ed. 45 LOEVE. Probability Theory I. 4th ed. 14 GOLUBITSKY/GUILLEMIN. Stable Mappings 46 LOEVE. Probability Theory II. 4th ed. and Their Singularities. 47 MoiSE. Geometric Topology in 15 BERBERIAN. Lectures in Functional Dimensions 2 and 3. Analysis and Operator Theory. 48 SACHS/WU. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERG/WEIR. Linear Geometry. 18 HALMOS. Measure Theory. 2nd ed. 19 HALMOS. A Hilbert Space Problem Book. 50 EDWARDS. Fermat's Last Theorem. 2nd ed. 51 KLINGENBERG. A Course in Differential 20 HuSEMOLLER. Fibre Bundles. 3rd ed. Geometry. 21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry. 22 BARNES/MACK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic. to Mathematical Logic. 54 GRAVER/WATKINS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs. 24 HOLMES. Geometric Functional Analysis 55 BROWN/PEARCY. Introduction to Operator and Its Applications. Theory I: Elements of Functional Analysis. 25 HEWITT/STROMBERG. Real and Abstract 56 MASSEY. Algebraic Topology: An Analysis. Introduction. 26 MANES. Algebraic Theories. 57 CROWELL/FOX. Introduction to Knot 27 KELLEY. General Topology. Theory. 28 ZARISKI/SAMUEL. Commutative Algebra. 5 8 KoBLiTZ. p-adic Numbers, p-adic Vol.1. Analysis, and Zeta-Functions. 2nd ed. 29 ZARISKI/SAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields. Vol.11. 60 ARNOLD. Mathematical Methods in 30 JACOBSON. Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed. Basic Concepts. 61 WHITEHEAD. Elements of Homotopy 31 JACOBSON. Lectures in Abstract Algebra II. Theory. Linear Algebra. 62 KARGAPOLOY/MERLZJAKOY. Fundamentals 32 JACOBSON. Lectures in Abstract Algebra of the Theory of Groups. III. Theory of Fields and Galois Theory. 63 BoLLOBAS. Graph Theory. 33 HiRSCH. Differential Topology. (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 [email protected] 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. Use in connection with any form of information storage and retrieval, electronic adaptation, com puter software, or by similar or dissimilar methodology now known or hereafter developed is for bidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (EB) 9 8 7 6 5 4 3 21 springeronline .com 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).
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