Function Classes on the Unit Disc (eBook)

An Introduction

Miroslav Pavlović (Autor)

eBook Download: PDF

2014
462 Seiten
De Gruyter (Verlag)
978-3-11-028190-3 (ISBN)

Infos

Themonograph contains a study on various function classes, a number of new results and new or easy proofs of old result (Fefferman-Stein theorem on subharmonic behavior,theorem on conjugate functions on Bergman spaces), which might be interesting for specialists, a full discussion on g-function (all p > 0), and a treatment of lacunary series with values in quasi-Banach spaces.

Miroslav Pavlovi?, University of Belgrade, Serbia.

Autor

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Miroslav Pavlović, University of Belgrade, Serbia.

Inhalt

Preface 7
1 The Poisson integral and Hardy spaces 15
1.1 The Poisson integral 19
1.1.1 Borel measures and the space h1 20
1.2 Spaces hp and Lp(T) (p > 1)
1.3 Space hp (p < 1)
1.4 Harmonic conjugates 32
1.4.1 Privalov–Plessner’s theorem and the Hilbert operator 33
1.5 Hardy spaces: basic properties 36
1.5.1 Radial limits and mean convergence 38
1.5.2 Space H1 41
1.6 Riesz projection theorem 43
1.6.1 Aleksandrov’s theorem 47
Further notes and results 49
2 Subharmonic functions and Hardy spaces 54
2.1 Basic properties of subharmonic functions 54
2.1.1 Maximum principle 56
2.2 Properties of the mean values 56
2.3 Riesz measure 59
2.3.1 Riesz’ representation formula 61
2.4 Factorization theorems 63
2.4.1 Inner–outer factorization 64
2.5 Some sharp inequalities 66
2.6 Hardy–Stein identities 72
2.6.1 Lacunary series 74
2.7 Subordination principle 75
2.7.1 Composition with inner functions 78
2.7.2 Approximation with inner functions 82
Further notes and results 83
3 Subharmonic behavior and mixed norm spaces 88
3.1 Quasi-nearly subharmonic functions 88
3.2 Regularly oscillating functions 89
3.3 Mixed norm spaces: definition and basic properties 97
3.4 Embedding theorems 106
3.5 Fractional integration 109
3.6 Weighted mixed norm spaces 113
3.6.1 Lacunary series in mixed norm spaces 116
3.6.2 Bergman spaces with rapidly decreasing weights 116
3.6.3 Mixed norm spaces with subnormal weights 119
3.7 Lq-integrability of lacunary power series 123
3.7.1 Lacunary series in C[0, 1] 126
Further notes and results 128
4 Taylor coefficients with applications 132
4.1 Using interpolation of operators on Hp 132
4.1.1 An embedding theorem 135
4.1.2 The case of monotone coefficients 140
4.2 Strong convergence in H1 143
4.2.1 Generalization to (C, a)-convergence 145
4.3 A (C, a)-maximal theorem 146
Further notes and results 149
5 Besov spaces 152
5.1 Decomposition of Besov spaces: case 1 < p <
5.2 Maximal function 154
5.3 Decomposition of Besov spaces: case 0 < p =8
5.3.1 Radial limits of Hardy–Bloch functions 159
5.4 Duality in the case 0 < p =8
5.5 Embeddings between Hardy and Besov spaces 169
5.6 Best approximation by polynomials 174
5.7 Normal Besov spaces 176
5.8 Inner functions in Besov and Hardy–Sobolev spaces 178
5.8.1 Approximation of a singular inner function 178
5.8.2 Hardy–Sobolev space Sp 1/p 184
5.8.3 f-property and K-property 185
Further notes and results 186
6 The dual of H1 and some related spaces 189
6.1 Norms on BMOA 189
6.2 Garsia’s and Fefferman’s theorems 193
6.2.1 Fefferman’s duality theorem 197
6.3 Vanishing mean oscillation 197
6.4 BMOA and Bp 1/p 199
6.4.1 Tauberian nature of Bp 1/p 202
6.5 Coefficients of BMOA functions 203
6.6 Bloch space 203
6.7 Mean growth of Hp-Bloch functions 206
6.8 Composition operators on B and BMOA 208
6.8.1 Weighted Bloch spaces 211
6.9 Proof of the bi-Bloch lemma 216
Further notes and results 220
7 Littlewood–Paley theory 225
7.1 Vector maximal theorems and Calderon’s area theorem 225
7.2 Littlewood–Paley g-theorem 227
7.3 Applications of the (C,m)-maximal theorem 231
7.4 Generalization of the ..-theorem 236
7.5 Proof of Calderón’s theorem 238
7.6 Littlewood–Paley inequalities 243
7.7 Hyperbolic Hardy classes 249
Further notes and results 252
8 Lipschitz spaces of first order 255
8.1 Definitions and basic properties 255
8.1.1 Lipschitz spaces of analytic functions 260
8.1.2 Mean Lipschitz spaces 261
8.2 Lipschitz condition for the modulus 263
8.3 Composition operators 265
8.4 Composition operators into H. p. 268
8.5 Inner functions 274
Further notes and results 275
9 Lipschitz spaces of higher order 278
9.1 Moduli of smoothness and related spaces 278
9.2 Lipschitz spaces and spaces of harmonic functions 281
9.3 Conjugate functions 289
9.4 Integrated mean Lipschitz spaces 292
9.4.1 Generalized Lipschitz spaces 294
9.5 Invariant Besov spaces 298
9.6 BMO-type characterizations of Lipschitz spaces 300
9.6.1 Division and multiplication by inner functions 304
Further notes and results 305
10 One-to-one mappings 308
10.1 Integral means of univalent functions 308
10.1.1 Distortion theorems 309
10.2 Membership of univalent functions in some function classes 312
10.3 Quasiconformal harmonic mappings 318
10.3.1 Boundary behavior of QCH homeomorphisms of the disk 318
10.4 Hp-classes of quasiconformal mappings 326
Further notes and results 329
11 Coefficients multipliers 332
11.1 Multipliers on abstract spaces 332
11.1.1 Compact multipliers 337
11.2 Multipliers for Hardy and Bergman spaces 338
11.2.1 Multipliers from H1 to BMOA 341
11.3 Solid spaces 343
11.3.1 Solid hull of Hardy spaces (0 < p <
11.4 Multipliers between Besov spaces 346
11.4.1 Monotone multipliers 349
11.5 Multipliers of spaces with subnormal weights 351
11.6 Some applications to composition operators 362
Further notes and results 363
12 Toward a theory of vector-valued spaces 366
12.1 Some properties of admissible spaces 366
12.2 Subharmonic behavior of ||F(z)||x 373
12.2.1 Banach envelope of Hp(X), 0 < p <
12.3 Linear operators on Hardy and Bergman spaces 378
12.4 Proof of the Coifman–Rochberg theorem 383
Further notes and results 388
A Quasi-Banach spaces 389
A.1 Quasi-Banach spaces 389
A.2 q-Banach envelopes 390
A.3 Closed graph theorem 393
A.4 F-spaces 396
A.4.1 Nevanlinna class 396
A.5 Spaces lp 397
A.6 Lacunary series in quasi-Banach spaces 398
A.6.1 Lp-integrability of lacunary series on (0, 1) 399
Further notes and results 409
B Interpolation and maximal functions 411
B.1 Riesz–Thorin theorem 411
B.2 Weak Lp-spaces and Marcinkiewicz’s theorem 413
B.3 Classical maximal functions 417
B.4 Rademacher functions and Khintchin’s inequality 423
B.5 Nikishin’s theorem 424
B.6 Nikishin–Stein’s theorem 426
B.7 Banach’s principle and the theorem on a.e. convergence 429
B.8 Vector-valued maximal theorem 431
Further notes and results 432
Bibliography 435
Index 457

Rezensionen

lt;P>"In this ambitious book, the author treats a number of topics from the theory of functions and function spaces on the unit disc in the complex plane. The selection of topics is far ranging, and includes both classical and modern ideas. Many of his proofs are new or unusual, and many of his ideas and presentations appear here in book form for the first time." — Steven G. Krantz, Mathematical Reviews

"[...] this is a well-written and detailed text with concise proofs. Graduate students and researchers who are pursuing research in harmonic or holomorphic function theory of one or several variables will find this book to be an excellent addition to their personal library." — Manfred Stoll, Zentralblatt für Mathematik

Produkt-Details

• Erscheint lt. Verlag: 12.12.2014
• Reihe/Serie: De Gruyter Studies in Mathematics
• Reihe/Serie: De Gruyter Studies in Mathematics
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Schulbuch / Wörterbuch ► Lexikon / Chroniken
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Bergman space • Bergman spaces • Besov-Lipschitz Space • bounded mean oscillation • Hardy space • Hardy Spaces • Harmonic Functions • H^p spaces • Littlewood-Paley g-Function • L^p spaces • Quasinormed spaces
• ISBN-10: 3-11-028190-2 / 3110281902
• ISBN-13: 978-3-11-028190-3 / 9783110281903