Bounded Analytic Functions

470 Pages·2007·2.017 MB·English

by  John B. Garnett (auth.)

236
Graduate Texts in Mathematics
Editorial Board
S.Axler K.A.Ribet
Graduate Texts in Mathematics
1 TAKEUTI/ZARING.Introduction to 34 SPITZER.Principles ofRandom 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.
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4 HILTON/STAMMBACH.A Course in Topological Spaces.
Homological Algebra.2nd ed. 37 MONK.Mathematical Logic.
5 MACLANE.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 41 APOSTOL.Modular Functions and
Algebras and Representation Theory. Dirichlet Series in Number Theory.
10 COHEN.A Course in Simple Homotopy 2nd ed.
Theory. 42 J.-P.SERRE.Linear Representations of
11 CONWAY.Functions ofOne Complex Finite Groups.
Variable I.2nd ed. 43 GILLMAN/JERISON.Rings of
12 BEALS.Advanced Mathematical Analysis. Continuous Functions.
13 ANDERSON/FULLER.Rings and 44 KENDIG.Elementary Algebraic
Categories ofModules.2nd ed. Geometry.
14 GOLUBITSKY/GUILLEMIN.Stable 45 LOÈVE.Probability Theory I.4th ed.
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15 BERBERIAN.Lectures in Functional 47 MOISE.Geometric Topology in
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16 WINTER.The Structure ofFields. 48 SACHS/WU.General Relativity for
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18 HALMOS.Measure Theory. 49 GRUENBERG/WEIR.Linear Geometry.
19 HALMOS.A Hilbert Space Problem 2nd ed.
Book.2nd ed. 50 EDWARDS.Fermat's Last Theorem.
20 HUSEMOLLER.Fibre Bundles.3rd ed. 51 KLINGENBERG.A Course in Differential
21 HUMPHREYS.Linear Algebraic Groups. Geometry.
22 BARNES/MACK.An Algebraic 52 HARTSHORNE.Algebraic Geometry.
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23 GREUB.Linear Algebra.4th ed. 54 GRAVER/WATKINS.Combinatorics with
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25 HEWITT/STROMBERG.Real and Abstract Operator Theory I:Elements of
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30 JACOBSON.Lectures in Abstract Algebra 59 LANG.Cyclotomic Fields.
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32 JACOBSON.Lectures in Abstract Algebra Theory.
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Theory. Fundamentals ofthe Theory ofGroups.
33 HIRSCH.Differential Topology. 63 BOLLOBAS.Graph Theory.
(continued after index)
John B. Garnett
Bounded Analytic
Functions
Revised First Edition
JohnB.Garnett
DepartmentofMathematics
6363MathematicalSciences
UniversityofCalifornia,LosAngeles
LosAngeles,CA.90095–1555
USA
[email protected]
EditorialBoard:
S.Axler K.A.Ribet
DepartmentofMathematics DepratmentofMathematics
SanFranciscoStateUniversity UniversityofCalifornia,Berkeley
SanFrancisco,CA94132 Berkeley,CA94720-3840
USA USA
[email protected] [email protected]
PreviouslypublishedbyAcademicPress,Inc.
SanDiego,CA92101
Mathematics:SubjectClassification(2000):30D5530–0246315
LibraryofCongressControlNumber:2006926458
ISBN-13:978-0387-33621-3
ISBN-10:0-387-33621-4
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To Dolores
Preface to Revised First Edition
ThiseditionofBoundedAnalyticFunctionsisthesameasthefirsteditionexcept
forthecorrectionsofseveralmathematicalandtypographicalerrors.Ithankthe
many colleagues and students who have pointed out errors in the first edition.
TheseincludeS.Axler,C.Bishop,A.Carbery,K.Dyakonov,J.Handy,V.Havin,H.
Hunziker,P.Koosis,D.Lubinsky,D.Marshall,R.Mortini,A.Nicolau,M.O’Neill,
W.Rudin,D.Sarason,D.Sua´rez,C.Sundberg,C.Thiele,S.Treil,I.Uriarte-Tuero,
J.Va¨isa¨la¨,N.Varopoulos,andL.Ward.
Ihadplannedtoprepareasecondeditionwithanupdatedbibliographyandan
appendixonresultsnewinthefieldsince1981,butthatworkhasbeenpostponedfor
toolong.Inthemeantimeseveralexcellentrelatedbookshaveappeared,including
M. Andersson, Topics in Complex Analysis; G. David and S. Semmes, Singular
Integrals and Rectifiable Sets in (cid:2)n and Analysis of and on Uniformly Rectifi-
ableSets;S.Fischer,Functiontheoryonplanardomains;P.Koosis,Introduction
to H spaces, Second edition; N. Nikolski, Operators, Functions, and Systems;
p
K.Seip,InterpolationandSamplinginSpacesofAnalyticFunctions;andB.Simon,
OrthogonalPolynomialsontheUnitCircle.
Severalproblemsposedinthefirsteditionhavebeensolved.Igivereferences
only to Mathematical Reviews. The question page 167 on when E∞ contains a
BlaschkeproductwassettledbyA.StrayinMR0940287.M.Papadimitrakis,MR
0947674,gaveacounterexampletotheconjectureinProblem5page170.Thelate
T.Wolff,MR1979771,hadacounterexampletotheBaernsteinconjecturecited
on page 260. S. Treil resolved the g2 problem on page 319 in MR 1945294. A
constructiveFefferman-SteindecompositionoffunctionsinBMO((cid:2)n)wasgiven
bythelateA.UchiyamainMR1007515,andC.Sundberg,MR0660188,found
aconstructiveproofoftheChang-Marshalltheorem.Problem5.3page420was
resolvedbyGarnettandNicolau,MR1394402,usingworkofMarshallandStray
MR1394401.Problem5.4.onpage420remainsapuzzle,butHjelleandNicolau
(PacificJournalofMathematics,2006)haveaninterestingresultonapproximation
of moduli. P. Jones, MR 0697611, gave a construction of the P. Beurling linear
operatorofinterpolation.
IthankSpringerandF.W.Gehringforpublishingthisedition.
JohnGarnett
vii
Contents
Preface vii
ListofSymbols xiii
I. PRELIMINARIES
1. Schwarz’sLemma 1
2. Pick’sTheorem 5
3. PoissonIntegrals 10
4. Hardy–LittlewoodMaximalFunction 19
5. NontangentialMaximalFunctionandFatou’sTheorem 27
6. SubharmonicFunctions 32
Notes 38
ExercisesandFurtherResults 39
II. HP SPACES
1. Definitions 48
2. BlaschkeProducts 51
3. Maxim(cid:2)alFunctionsandBoundaryValues 55
4. (1/π) (log|f(t)|/(1+t2))dt >−∞ 61
5. TheNevanlinnaClass 66
6. InnerFunctions 71
7. Beurling’sTheorem 78
Notes 83
ExercisesandFurtherResults 84
III. CONJUGATEFUNCTIONS
1. Preliminaries 98
2. The Lp Theorems 106
3. ConjugateFunctionsandMaximalFunctions 111
ix
x contents
Notes 118
ExercisesandFurtherResults 120
IV. SOMEEXTREMALPROBLEMS
1. DualExtremalProblems 128
2. TheCarleson–JacobsTheorem 134
3. TheHelson–Szego¨ Theorem 139
4. InterpolatingFunctionsofConstantModulus 145
5. ParametrizationofK 151
6. Nevanlinna’sProof 159
Notes 168
ExercisesandFurtherResults 168
V. SOMEUNIFORMALGEBRA
1. MaximalIdealSpaces 176
2. InnerFunctions 186
3. AnalyticDiscsinFibers 191
4. RepresentingMeasuresandOrthogonalMeasures 193
5. TheSpace L1/H1 198
0
Notes 205
ExercisesandFurtherResults 206
VI. BOUNDEDMEANOSCILLATION
1. Preliminaries 215
2. TheJohn–NirenbergTheorem 223
3. Littlewood–PaleyIntegralsandCarlesonMeasures 228
4. Fefferman’sDualityTheorem 234
5. VanishingMeanOscillation 242
6. WeightedNormInequalities 245
Notes 259
ExercisesandFurtherResults 261
VII. INTERPOLATINGSEQUENCES
1. Carleson’sInterpolationTheorem 275
2. TheLinearOperatorofInterpolation 285
3. Generations 289
4. HarmonicInterpolation 292
5. Earl’sElementaryProof 299
Notes 304
ExercisesandFurtherResults 305
contents xi
VIII. THECORONACONSTRUCTION
1. InhomogeneousCauchy–RiemannEquations 309
2. TheCoronaTheorem 314
3. TwoTheoremsonMinimumModulus 323
4. InterpolatingBlaschkeProducts 327
5. Carleson’sConstruction 333
6. GradientsofBoundedHarmonicFunctions 338
7. AConstructiveSolutionof∂b/∂z¯ =μ 349
Notes 357
ExercisesandFurtherResults 358
IX. DOUGLASALGEBRAS
1. TheDouglasProblem 364
2. H∞+C 367
3. TheChang-MarshallTheorem 369
4. TheStructureofDouglasAlgebras 375
5. TheLocalFatouTheoremandanApplication 379
Notes 384
ExercisesandFurtherResults 385
X. INTERPOLATINGSEQUENCESANDMAXIMALIDEALS
1. AnalyticDiscsinM 391
2. Hoffman’sTheorem 401
3. ApproximateDependencebetweenKernels 406
4. InterpolatingSequencesandHarmonicSeparation 412
5. AConstructiveDouglas–RudinTheorem 418
Notes 428
ExercisesandFurtherResults 428
BIBLIOGRAPHY 434
INDEX 453
List of Symbols
Symbol Page Symbol Page
A 120 |E|=Lebesguemeasureof E 23
o
Am 121 E 292
A∞ 121 f(m) 180
Aα 57 fˆ(m)=Gelfandtransform 178
A−1 176 fˆ(s)=Fouriertransform 59
Aˆ 179 f (t) 15
x
A⊥ 196 f+(x) 324
A 57 f#(x) 271
N
B 237 f∗(t) 55
B 274 (f˜)∗(θ) 106
0
BLO 271 G 401
BMO 216 Gδ(b) 371
BMOA 262 Hf(θ) 102
BMO(T) 218 Hf(x) 105
BMOd 266 Hεf(x) 123
BUC 242 H∗f(x) 123
[B,H] 270 Hp = Hp(D) 48
B ={f analytic,|f(z)|≤1} 7 Hp = Hp(dt) 49
B,asetofinnerfunctions 364 H∞ 48
B 364 Hq 128
A 0
C =C(T) 128 H1 267
d
COP 432 H1 236
R
C 367 (H∞)−1 63
A
Cl(f,z) 76 H∞+C 132
D 1 [H∞,B¯] 364
D¯ 1 [H∞,U ] 365
A
dist(f,Hp) 129 H 5
dist∗ 241 I˜ 221
D 391 J 404
m
∂/∂z 312 J(f , f ,... , f ) 319
1 2 n
∂/∂z¯ 309 K(z ,r) 2
0
∂¯ 354 Kδ(B) 401
E∗ 323 LC(x) 235
xiii