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Garnett J.B. — Bounded Analytic Functions
Garnett J.B. — Bounded Analytic Functions

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Название: Bounded Analytic Functions

Автор: Garnett J.B.

Аннотация:

This book is an account of the theory of Hardy spaces in one dimension, with emphasis on some of the exciting developments of the past two decades or so. The last seven of the ten chapters are devoted in the main to these recent developments. The motif of the theory of Hardy spaces is the interplay between real, complex, and abstract analysis. While paying proper attention to each of the three aspects, the author has underscored the effectiveness of the methods coming from real analysis, many of them developed as part of a program to extend the theory to Euclidean spaces, where the complex methods are not available...Each chapter ends with a section called Notes and another called Exercises and further results. The former sections contain brief historical comments and direct the reader to the original sources for the material in the text.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second Edition

Год издания: 2006

Количество страниц: 459

Добавлена в каталог: 19.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(A_1)$ condition      272
$(A_2)$ condition      250
$(A_p)$ condition      247
$(A_{\infty})$ condition      256
$A_0$      see Disc algebra
$A_{\alpha}$      see Lipschitz class
$C_A$      367 376 378
$H^1$      48
$H^1$ and absolute continuity      59
$H^1$ and bounded variation      89
$H^1$ and cosets of $L^{\infty}/H^{\infty}$      147 154 198
$H^1$ as real Banach space      235 see
$H^1$, arguments of functions in      153 172
$H^1$, atomic decomposition      266
$H^1$, dual space of      238 362
$H^1$, extreme points of ball      151
$H^1_{\mathbb{R}}$      236 266
$H^2$      48 79 see
$H^p$      48—51 128
$H^p$ and extremal problems      127—134
$H^p$ and harmonic majorants      58
$H^p$ and interpolating sequences      307
$H^p$ and maximal functions      55 115
$H^p$ as sums of outer functions      84
$H^p$, boundary behavior      55—61 185
$H^p$, derivatives from      86
$H^p$, distance to      129
$H^p$, dual spaces      87 235 238
$H^p$, factorizations in      53 71 84
$H^p$, Fourier series of      56
$H^p$, Fourier transform of      59
$H^p$, real or positive functions in      91—93
$H^{\infty} + C$      132 172 367—369 384 386
$H^{\infty} + C$, invertible functions in      385
$H^{\infty} + C$, maximal ideals of      368
$H^{\infty}$      48 63 364
$H^{\infty}$ and Banach approximation property      205
$H^{\infty}$ as logmodular algebra      63 194
$H^{\infty}$ as uniform algebra      183—191
$H^{\infty}$, analytic structure in maximal ideal space      191—193 391—406
$H^{\infty}$, best approximation in      130 134 168 170—171
$H^{\infty}$, cosets of in $L^{\infty}/H^{\infty}$      146 154
$H^{\infty}$, distance to      130 131 142 241 363 376
$H^{\infty}$, exposed points of      213
$H^{\infty}$, extreme points of      173
$H^{\infty}$, ideals of      82 83 183 314 319 359
$H^{\infty}$, interpolations by      see Interpolating functions; Interpolating sequences; Interpolation problem
$H^{\infty}$, maximal ideal space      181—182 314 391—395 419
$H^{\infty}$, measures orthogonal to      131 196 198 212
$H^{\infty}$, peak sets of      199
$H^{\infty}$, predual of      199—205 212
$H^{\infty}$, representing measures for      193—198 211 365
$H^{\infty}$, Silov boundary of      184
$H^{\infty}$, weak-star closed ideals      82
$H^{\infty}$, weak-star density of polynomials      85
$H^{\infty}$, weakly continuous functionals on      199—205 212
$l^p$      12 14 31 45 56 123 128
$L^p$ and conjugate functions      104 106—108
$L^p$ and interpolating sequences      307
$L^p$ and representing measures      195
$l^{\infty}$      180
$L^{\infty}$ and conjugate functions      101 110 139 220
$l^{\infty}$, complemented in $H^{\infty}$      285
$L^{\infty}$, distance to      241 250
$l^{\infty}$, interpolation in      276
$N^+$      68 71 94
$Q_A$      367 375 378
Adamyan, Arov, and Krein theorem      146 154
Amar — Lederer lemma      199
Analytic discs      192 392 428
Analytic discs and Carleson measures      429
Analytic discs and interpolating sequences      399 404
Analytic map      192 392 405 431
Ando’s theorem      203 205
Angle between subspaces      141 171
Angular derivative      42 432
Approximate identity      19
atom      266
B      see Bloch functions
B = ball $H^{\infty}$      1 9 10 39 41 151
B = set of inner functions      364
Bad function      125
Badly approximable function      170 173
Balyage      221
Banach algebra      176
Banach approximation property      205
Banach — Alaoglu theorem      18
Bernard’s theorem      189
Best approximation      130
Best approximation in $H^{\infty} + C$      386
Best approximation in $H^{\infty}$      130 134 168 170 171
Best approximation in Douglas algebra      386
Best approximation, continuity of      134 170
Beurling, P., theorem of      285 304
Beurling’s theorem      78
Blaschke condition      51
Blaschke products      51 52 75 76
Blaschke products and Silov boundary      189
Blaschke products as interpolating functions      9 167 305
Blaschke products, boundary continuity      72
Blaschke products, characterization      54
Blaschke products, density in inner functions      76
Blaschke products, factorizations      397 401
Blaschke products, finite      5 9 132 133 145 173 302 420
Blaschke products, interpolating      see Interpolating Blaschke product
Blaschke products, logarithms of moduli of      279
Bloch functions      273 274 432
BMO      216—228 237—242 261—274
BMO and $(A_p)$ weights      249
BMO and Carleson measure      222 233 264 265 267 318
BMO and conformal mappings      273
BMO and conjugate functions      227 240 349
BMO and interpolating sequences      307
BMO and Littlewood — Paley integrals      233 262
BMO as dual of $H^1_{\mathbb{R}}$      238 267 362
BMO, distance to $L^{\infty}$ in      250 273
BMO, dyadic      266
BMOA      266 273—274
Boundary      181
Boundary, Silov      see Silov boundary
Bounded lower oscillation      272
Bounded mean oscillation      see BMO
Box kernel      22 217
Burkholder, Gundy, and Silverstein theorem      111
Calderon — Zygmund lemma      223
Canonical Factorization theorem      71
Capacity      75 273
Caratheodory theorem      6 89
Carleson measures      30 61 231
Carleson measures and analytic disc      429
Carleson measures and arc length      330
Carleson measures and BMO      221 233 259 264 267 318
Carleson measures and Cauchy — Rieman equations      311 312 349
Carleson measures and conformal invariance      231
Carleson measures and gradients      233 259 264 267
Carleson measures and interpolating sequences      278 349
Carleson measures and Littlewood — Paley integrals      233 259
Carleson norm      30
Carleson — Jacobs theorem      134
Carleson, theorems of      31 61 120 122 185 278 315
Cauchy — Riemann equations, inhomogeneous      309
Chang — Marshall theorem      364 369 389
Chebychev inequality      20
Cluster set      76
Cluster set and fiber      209
Commutator      270
Complemented subspace      286
Complex homomorphism      176
Cone $\Gamma_{\alpha} (e^{I \varphi})$      23
Cone $\Gamma_{\alpha} (t)$      21
Conformal mappings and Bloch functions      273
Conformal mappings and BMO      273
Conformal mappings, boundary behavior      89
Conjugate functions      63 98 378
Conjugate functions and $L^1$      106 111
Conjugate functions and $L^p$      108
Conjugate functions and $L^{\infty}$      101 109 141 220
Conjugate functions and BMO      220 227 240
Conjugate functions and maximal functions      111
Conjugate functions and VMO      245 see
Conjugate functions in $L^2 (\mu)$      144
Conjugate functions in $L^p (\mu)$      247
Conjugate functions, continuity of      101 122
Conjugate functions, maximal      106
Conjugate Poisson kernel      99 105
Conjugation operator      99
Constant of interpolation      276 284
Convolution      11 13
Corona      314
Corona data      315
Corona problem on plane domains      358—359
Corona solutions      315
Corona theorem      185 315
Cosets of $L^{\infty} /H^{\infty}$      146 151
Covering lemma      24
de Leeuw — Rudin theorem      151 153
Degree of finite Blaschke product      6
Derivatives of $H^p$ functions      86 265 see
Dilation      13
Dini continuity      101 122 134
Dirichlet algebra      194 210
Disc algebra      120 128 131 171 183 185 189 207 219 366 384 385
Disc, euclidean      3
Disc, noneuclidean      2
Distance in $L^2 (\mu)$      139 171
Distance in BMO to $L^{\infty}$      241 250
Distance to $A_0$      131
Distance to $H^p$      129 168
Distance to $H^{\infty} + C$      385 386
Distance to $H^{\infty}$      130 131 142 241 363 376
Distance to $Re H^{\infty}$      241 251
Distribution function      20
Douglas algebras      364 369
Douglas algebras and VMO      375
Douglas algebras, characterizations      371 382
Douglas algebras, maximal ideal spaces      365 366
Douglas algebras, structure of      375—379
Douglas problem      364
Douglas — Rudin theorem      192 209 365 389
Douglas — Rudin theorem, constructive      418
Dual extremal function      129 130 131
Dual extremal problem      128 169 282
Dual space of $H^1$      238
Dual space of $H^p$      87 234
Dual space of VMO      268
Dyadic $H^1$      267
Dyadic BMO      266
Dyadic maximal function      267
Earl’s construction      299
Endomorphism of $H^{\infty}$      430
Expectation      293
Exposed point      153 213
Extremal function      134 146 154 166 169 see
Extremal problem, dual      128 169 282
Extremal problem, nonlinear      287
Extreme point of ball $(H^1)$      151
Extreme point of ball $(H^{\infty})$      173
Extremely disconnected space      207
Fatou theorem      28 120 339
Fatou theorem, local      90 380
Fefferman — Stein theorems      119 271
Fefferman’s theorem      238
Fefferman’s theorem, constructively      349
Fiber      183 207 385
Fiber algebra      385
Fiber and cluster set      209
Fourier multiplier      104
Fourier series      103
Fourier series of $H^p$ function      56
Fourier transform      59 106
Frostman theorem      75
Gehring’s inequality      252
Gelfand topology      178
Gelfand transform      178
Generates      189
Generations      289 415 416
Generations and corona construction      335
Generations and dependencies of kernals      408
Generations and gradients of functions      341
Generations and interpolating Blaschke products      329 see
Geodesic, hyperbolic      5 39 40
Gleason distance      40
Gleason parts      393
Gleason parts and analytic discs      393 399
Gleason parts, homeomorphic to unit disc      430
Gleason parts, one point      393 404 429
Good $\lambda$ inequality      258 260
Good function      125
Gradients of harmonic functions      228 233 338 362
Gradients of harmonic functions and BMO      262 264 see
Greatest common divisor (of family of inner functions)      80
Greatest lower bound (of family of measures)      80
Green’s potential      75 94
Green’s Theorem      228
Hall’s lemma      357 361
Hardy space      see $H^p$
Hardy — Littlewood inequality      116
Hardy — Littlewood maximal functions      21 23 246 327
Hardy — Littlewood maximal theorem      23
Hardy’s inequality      89 264
Harmonic conjugate      63 see
Harmonic function      10
Harmonic interpolating sequence      293 304 361 411
Harmonic majorant      36
Harmonic measure      12 40 357
Helson — Szego theorem      141 246 250
Herglotz’s theorem      19
Hilbert transform      101 123 141 144 246 256 378 385 see
Hoffman’s theorem      401 404
Homogeneity of kernels      13
Homomorphism, complex      176
Hunt, Muckenhoupt, and Wheeden theorem      247 260
Ideal, generated by $\{f_1, \cdots, f_n \}$      319 359
Ideal, maximal      177
Ideal, square of      404
Ideal, weak-star closed      82
Inhomogeneous Cauchy — Riemann equations      309
Inhomogeneous Cauchy — Riemann equations and interpolating Blaschke products      349
Inner factor      74
Inner functions      71 75 76
Inner functions and $L^{\infty}$      186
Inner functions and Douglas algebras      369
Inner functions and ideals in $H^{\infty}$      82
Inner functions and invariant subspaces      79
Inner functions and Silov boundary      187
Inner functions as interpolating functions      145 166
Inner functions, approximation by Blaschke product      76
Inner functions, behavior of singularities      77
Inner functions, generating $H^{\infty}$      189
Integrable, locally      215
Interpolating Blaschke products      327 395
Interpolating Blaschke products and approximation problem      420
Interpolating Blaschke products and Douglas algebras      369 370 388
Interpolating Blaschke products and inhomogeneous Cauchy — Riemann equations      349
Interpolating Blaschke products and maximal ideal space      419
Interpolating Blaschke products and nontangential points      428
Interpolating Blaschke products and Silov boundary      328
Interpolating Blaschke products as interpolating functions      300
Interpolating Blaschke products, factorization of      397
Interpolating Blaschke products, maximal ideals containing      370
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