Hardy classes on infinitely connected Riemann surfaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1983
|
| Schriftenreihe: | Lecture notes in mathematics
1027 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 280 S. |
| ISBN: | 3540127291 0387127291 |
Internformat
MARC
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| 100 | 1 | |a Hasumi, Morisuke |d 1932- |e Verfasser |0 (DE-588)1082328898 |4 aut | |
| 245 | 1 | 0 | |a Hardy classes on infinitely connected Riemann surfaces |c Morisuke Hasumi |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1983 | |
| 300 | |a XII, 280 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Lecture notes in mathematics |v 1027 | |
| 650 | 4 | |a Hardy, Espaces de | |
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| 650 | 4 | |a Hardy classes | |
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Datensatz im Suchindex
| _version_ | 1804114691185704960 |
|---|---|
| adam_text | CONTENTS
PREFACE iii
CHAPTER I. THEORY OF RIEMANN SURFACES: A QUICK REVIEW
51. Topology of Riemann Surfaces 1
1. Exhaustion 1
2. The Homology Groups 2
3. The Fundamental Group 3
§2. Classical Potential Theory 4
4. Superharmonic Functions 4
5. The Dirichlet Problem 5
6. Potential Theory 7
§3. Differentials 9
7. Basic Definition 9
8. The Class r and its Subclasses 11
9. Cycles and Differentials 12
10. Riemann Roch Theorem 14
11. Cauchy Kernels on Compact Bordered Surfaces 17
Notes 22
CHAPTER II. MULTIPLICATIVE ANALYTIC FUNCTIONS
51. Multiplicative Analytic Functions 23
1. The First Cohomology Group 23
2. Line Bundles and Multiplicative Analytic Functions ... 28
3. Existence of Holomorphic Sections 31
§2. Lattice Structure of Harmonic Functions 33
4. Basic Structure 33
5. Orthogonal Decomposition 36
Notes 38
CHAPTER III. MARTIN COMPACTIFICATION
51. Compactification 39
1. Definition 39
2. Integral Representation 40
3. The Dirichlet Problem 43
X
§2. Fine Limits 49
4. Definition of Fine Limits 49
5. Analysis of Boundary Behavior 50
§3. Covering Maps 57
6. Correspondence of Harmonic Functions 57
7. Preservation of Harmonic Measures 59
Notes 63
CHAPTER IV. HARDY CLASSES
§1. Hardy Classes on the Unit Disk 64
1. Basic Definitions 64
2. Some Classical Results 66
§2. Hardy Classes on Hyperbolic Riemann Surfaces 73
3. Boundary Behavior of H^ and h^ Functions 73
4. Some Results on Multiplicative Analytic Functions ... 74
5. The g Topology 75
Notes 82
CHAPTER V. RIEMANN SURFACES OF PARREAU WIDOM TYPE
§1. Definitions and Fundamental Properties 83
1. Basic Definitions 83
2. Widom s Characterization 85
3. Regularization of Surfaces of Parreau Widom Type .... 86
§2. Proof of Widom s Theorem (I) 90
4. Analysis on Regular Subregions 90
5. Proof of Necessity 95
§3. Proof of Widom s Theorem (II) 99
6. Review of Principal Operators 99
7. Modified Green Functions 102
8. Proof of Sufficiency 111
9. A Few Direct Consequences 117
Notes 118
CHAPTER VI. GREEN LINES
§1. The Dirichlet Problem on the Space of Green Lines 119
1. Definition of Green Lines 119
2. The Dirichlet Problem 121
§2. The Space of Green Lines on a Surface of Parreau Widom Type 124
3. The Green Star Regions 124
4. Limit along Green Lines 129
XI
§3. The Green Lines and the Martin Boundary 132
5. Convergence of Green Lines 132
6. Green Lines and the Martin Boundary 135
7. Boundary Behavior of Analytic Maps 140
Notes 143
CHAPTER VII. CAUCHY THEOREMS
§1. The Inverse Cauchy Theorem 144
1. Statement of Results 144
2. Proof of Theorem IB 145
52. The Direct Cauchy Theorem 151
3. Formulation of the Condition 151
4. The Direct Cauchy Theorem of Weak Type 152
§3. Applications 155
5. Weak star Maximality of H 155
6. Common Inner Factors 156
7. The Orthocomplement of H°°(dx) 157
Notes 159
CHAPTER VIII. SHIFT INVARIANT SUBSPACES
§1. Preliminary Observations 160
1. Generalities 160
2. Shift Invariant Subspaces on the Unit Disk 162
§2. Invariant Subspaces 167
3. Doubly Invariant Subspaces 167
4. Simply Invariant Subspaces 169
5. Equivalence of (DCT ) 177
Notes 178
CHAPTER IX. CHARACTERIZATION OF SURFACES OF PARREAU WIDOM TYPE
§1. The Inverse Cauchy Theorem and Surfaces of Parreau Widom Type 179
1. Statement of the Main Result 179
2. A Mean Value Theorem 183
3. Proof of the Main Theorem 187
§2. Conditions Equivalent to the Direct Cauchy Theorem 19 8
4. General Discussion 198
5. Functions mp(£,a) and (DCT) 200
Notes 207
XII
CHAPTER X. EXAMPLES OF SURFACES OF PARREAU WIDOM TYPE
§1. PWS of Infinite Genus for Which (DCT) Holds 208
1. PWS s of Myrberg Type 208
2. Verification of (DCT) 213
§2. Plane Regions of Parreau Widom Type for Which (DCT) Fails . 215
3. Some Simple Lemmas 215
4. Existence Theorem 217
§3. Further Properties of PWS 2 21
5. Embedding into the Maximal Ideal Space 221
6. Density of H°°(R) 223
§4. The Corona Problem for PWS 227
7. (DCT) and the Corona Theorem: Positive Examples .... 227
8. Negative Examples 229
Notes 233
CHAPTER XI. CLASSIFICATION OF PLANE REGIONS
SI. Hardy Orlicz Classes 234
1. Definitions 234
2. Some Basic Properties 235
§2. Null Sets of Class N. 238
3. Preliminary Lemmas 238
4. Existence of Null Sets 247
§3. Classification of Plane Regions 253
5. Lemmas 253
6. Classification Theorem 256
7. Majoration by Quasibounded Harmonic Functions 260
Notes 261
APPENDICES
A.I. The Classical Fatou Theorem 262
A.2. Kolmogorov s Theorem on Conjugate Functions 267
A. 3. The F. and M. Riesz Theorem 269
References 272
Index of Notations 276
Index 278
|
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| indexdate | 2024-07-09T15:10:44Z |
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| isbn | 3540127291 0387127291 |
| language | English |
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| physical | XII, 280 S. |
| publishDate | 1983 |
| publishDateSearch | 1983 |
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| publisher | Springer |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Hasumi, Morisuke 1932- Verfasser (DE-588)1082328898 aut Hardy classes on infinitely connected Riemann surfaces Morisuke Hasumi Berlin [u.a.] Springer 1983 XII, 280 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1027 Hardy, Espaces de Riemann, Surfaces de Riemann-vlakken gtt Hardy classes Riemann surfaces Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Hardy-Klasse (DE-588)4159107-0 gnd rswk-swf Riemannsche Fläche (DE-588)4049991-1 s Hardy-Klasse (DE-588)4159107-0 s DE-604 Lecture notes in mathematics 1027 (DE-604)BV000676446 1027 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000136585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hasumi, Morisuke 1932- Hardy classes on infinitely connected Riemann surfaces Lecture notes in mathematics Hardy, Espaces de Riemann, Surfaces de Riemann-vlakken gtt Hardy classes Riemann surfaces Riemannsche Fläche (DE-588)4049991-1 gnd Hardy-Klasse (DE-588)4159107-0 gnd |
| subject_GND | (DE-588)4049991-1 (DE-588)4159107-0 |
| title | Hardy classes on infinitely connected Riemann surfaces |
| title_auth | Hardy classes on infinitely connected Riemann surfaces |
| title_exact_search | Hardy classes on infinitely connected Riemann surfaces |
| title_full | Hardy classes on infinitely connected Riemann surfaces Morisuke Hasumi |
| title_fullStr | Hardy classes on infinitely connected Riemann surfaces Morisuke Hasumi |
| title_full_unstemmed | Hardy classes on infinitely connected Riemann surfaces Morisuke Hasumi |
| title_short | Hardy classes on infinitely connected Riemann surfaces |
| title_sort | hardy classes on infinitely connected riemann surfaces |
| topic | Hardy, Espaces de Riemann, Surfaces de Riemann-vlakken gtt Hardy classes Riemann surfaces Riemannsche Fläche (DE-588)4049991-1 gnd Hardy-Klasse (DE-588)4159107-0 gnd |
| topic_facet | Hardy, Espaces de Riemann, Surfaces de Riemann-vlakken Hardy classes Riemann surfaces Riemannsche Fläche Hardy-Klasse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000136585&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT hasumimorisuke hardyclassesoninfinitelyconnectedriemannsurfaces |