Function classes on the unit disc: an introduction
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin
De Gruyter
2014
|
| Schriftenreihe: | De Gruyter Studies in Mathematics
52 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis Klappentext |
| Beschreibung: | XIII, 449 S. graph. Darst. 240 mm x 170 mm |
| ISBN: | 9783110281231 9783110281903 9783110281910 |
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| 245 | 1 | 0 | |a Function classes on the unit disc |b an introduction |c Miroslav Pavlović |
| 263 | |a 201312 | ||
| 264 | 1 | |a Berlin |b De Gruyter |c 2014 | |
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Datensatz im Suchindex
| _version_ | 1809769108499070976 |
|---|---|
| adam_text |
CONTENTS
PREFACE VII
1 THE POISSON INTEGRAL AND HARDY SPACES 1
1.1 THE POISSON INTEGRAL 5
1.1.1 BOREL MEASURES AND THE SPACE H
1
6
1.2 SPACES H
P
AND L
P
(T) (P 1) 10
1.3 SPACED
(P
1) 12
1.4 HARMONIC CONJUGATES 18
1.4.1 PRIVALOV-PLESSNER'S THEOREM AND THE HILBERT OPERATOR
1.5 HARDY SPACES: BASIC PROPERTIES 22
1.5.1 RADIAL LIMITS AND MEAN CONVERGENCE 24
1.5.2 SPACE H
1
27
1.6 RIESZ PROJECTION THEOREM 29
1.6.1 ALEKSANDROV'S THEOREM 33
FURTHER NOTES AND RESULTS 35
2 SUBHARMONIC FUNCTIONS AND HARDY SPACES 40
2.1 BASIC PROPERTIES OF SUBHARMONIC FUNCTIONS 40
2.1.1 MAXIMUM PRINCIPLE 42
2.2 PROPERTIES OF THE MEAN VALUES 42
2.3 RIESZ MEASURE 45
2.3.1 RIESZ' REPRESENTATION FORMULA 47
2.4 FACTORIZATION THEOREMS 49
2.4.1 INNER-OUTER FACTORIZATION 50
2.5 SOME SHARP INEQUALITIES 52
2.6 HARDY-STEIN IDENTITIES 58
2.6.1 LACUNARY SERIES 60
2.7 SUBORDINATION PRINCIPLE 61
2.7.1 COMPOSITION WITH INNER FUNCTIONS 64
2.7.2 APPROXIMATION WITH INNER FUNCTIONS 68
FURTHER NOTES AND RESULTS 69
3 SUBHARMONIC BEHAVIOR AND MIXED NORM SPACES 74
3.1 QUASI-NEARLY SUBHARMONIC FUNCTIONS 74
3.2 REGULARLY OSCILLATING FUNCTIONS 75
3.3 MIXED NORM SPACES: DEFINITION AND BASIC PROPERTIES-
3.4 EMBEDDING THEOREMS 92
3.5 FRACTIONAL INTEGRATION 95
HTTP://D-NB.INFO/1032801077
X
CONTENTS
3.6 WEIGHTED MIXED NORM SPACES 99
3.6.1 LACUNARY SERIES IN MIXED NORM SPACES 102
3.6.2 BERGMAN SPACES WITH RAPIDLY DECREASING WEIGHTS 102
3.6.3 MIXED NORM SPACES WITH SUBNORMAL WEIGHTS 105
3.7 L^-INTEGRABILITY OF LACUNARY POWER SERIES 109
3.7.1 LACUNARY SERIES IN C[0,1] 112
FURTHER NOTES AND RESULTS 114
4 TAYLOR COEFFICIENTS WITH APPLICATIONS 118
4.1 USING INTERPOLATION OF OPERATORS ON H
P
118
4.1.1 AN EMBEDDINGTHEOREM 121
4.1.2 THE CASE OF MONOTONE COEFFICIENTS 126
4.2 STRONG CONVERGENCE IN H
1
129
4.2.1 GENERALIZATION TO (C, A)-CONVERGENCE 131
4.3 A (C,A)-MAXIMAL THEOREM 132
FURTHER NOTES AND RESULTS 135
5 BESOV SPACES 138
5.1 DECOMPOSITION OF BESOV SPACES: CASE 1 P O 138
5.2 MAXIMAL FUNCTION 140
5.3 DECOMPOSITION OF BESOV SPACES: CASE 0 P OO 143
5.3.1 RADIAL LIMITS OF HARDY-BLOCH FUNCTIONS 145
5.4 DUALITY IN THE CASE 0 P OO 149
5.5 EMBEDDINGS BETWEEN HARDY AND BESOV SPACES 155
5.6 BEST APPROXIMATION BY POLYNOMIALS 160
5.7 NORMAL BESOV SPACES 162
5.8 INNER FUNCTIONS IN BESOV AND HARDY-SOBOLEV SPACES 164
5.8.1 APPROXIMATION OF A SINGULAR INNER FUNCTION 164
5.8.2 HARDY-SOBOLEV SPACE S
PLJP
170
5.8.3 F-PROPERTY AND K-PROPERTY 171
FURTHER NOTES AND RESULTS 172
6 THE DUAL OF H
1
AND SOME RELATED SPACES 175
6.1 NORMS ON BMOA 175
6.2 GARSIA'S AND FEFFERMAN'S THEOREMS 179
6.2.1 FEFFERMAN'S DUALITY THEOREM 183
6.3 VANISHING MEAN OSCILLATION 183
6.4 BMOA AND 185
6.4.1 TAUBERIAN NATURE OF 8F.* 188
I/P
6.5 COEFFICIENTS OF BMOA FUNCTIONS 189
6.6 BLOCH SPACE 189
CONTENTS
XI
6.7 MEAN GROWTH OF H
P
-BLOCH FUNCTIONS 192
6.8 COMPOSITION OPERATORS ON 23 AND BMOA 194
6.8.1 WEIGHTED BLOCH SPACES 197
6.9 PROOF OF THE BI-BLOCH LEMMA 202
FURTHER NOTES AND RESULTS 206
7 LITTLEWOOD-PALEY THEORY 211
7.1 VECTOR MAXIMAL THEOREMS AND CATDERON'S AREA THEOREM 211
7.2 LITTLEWOOD-PALEY ^-THEOREM 213
7.3 APPLICATIONS OF THE (C,M)-MAXIMAL THEOREM 217
7.4 GENERALIZATION OF THE ^-THEOREM 222
7.5 PROOF OF CALDERON'S THEOREM 224
7.6 LITTLEWOOD-PALEY INEQUALITIES 229
7.7 HYPERBOLIC HARDY CLASSES 235
FURTHER NOTES AND RESULTS 238
8 LIPSCHITZ SPACES OF FIRST ORDER 241
8.1 DEFINITIONS AND BASIC PROPERTIES 241
8.1.1 LIPSCHITZ SPACES OF ANALYTIC FUNCTIONS 246
8.1.2 MEAN LIPSCHITZ SPACES 247
8.2 LIPSCHITZ CONDITION FOR THE MODULUS 249
8.3 COMPOSITION OPERATORS 251
8.4 COMPOSITION OPERATORS INTO HA
PA
254
8.5 INNER FUNCTIONS 260
FURTHER NOTES AND RESULTS 261
9 LIPSCHITZ SPACES OF HIGHER ORDER 264
9.1 MODULI OF SMOOTHNESS AND RELATED SPACES 264
9.2 LIPSCHITZ SPACES AND SPACES OF HARMONIC FUNCTIONS 267
9.3 CONJUGATE FUNCTIONS 275
9.4 INTEGRATED MEAN LIPSCHITZ SPACES 278
9.4.1 GENERALIZED LIPSCHITZ SPACES 280
9.5 INVARIANT BESOV SPACES 284
9.6 BMO-TYPE CHARACTERIZATIONS OF LIPSCHITZ SPACES 286
9.6.1 DIVISION AND MULTIPLICATION BY INNER FUNCTIONS 290
FURTHER NOTES AND RESULTS 291
10 ONE-TO-ONE MAPPINGS 294
10.1 INTEGRAL MEANS OF UNIVALENT FUNCTIONS 294
10.1.1 DISTORTION THEOREMS 295
10.2 MEMBERSHIP OF UNIVALENT FUNCTIONS IN SOME FUNCTION CLASSES 298
XII
CONTENTS
10.3 QUASICONFORMAL HARMONIC MAPPINGS 304
10.3.1 BOUNDARY BEHAVIOR OF QCH HOMEOMORPHISMS OF THE DISK 304
10.4 H
P
-CLASSES OF QUASICONFORMAL MAPPINGS 312
FURTHER NOTES AND RESULTS 315
11 COEFFICIENTS MULTIPLIERS 318
11.1 MULTIPLIERS ON ABSTRACT SPACES 318
11.1.1 COMPACT MULTIPLIERS 323
11.2 MULTIPLIERS FOR HARDY AND BERGMAN SPACES 324
11.2.1 MULTIPLIERS FROM H
1
TO BMOA 327
11.3 SOLID SPACES 329
11.3.1 SOLID HULL OF HARDY SPACES (0 P 1) 331
11.4 MULTIPLIERS BETWEEN BESOV SPACES 332
11.4.1 MONOTONE MULTIPLIERS 335
11.5 MULTIPLIERS OF SPACES WITH SUBNORMAL WEIGHTS 337
11.6 SOME APPLICATIONS TO COMPOSITION OPERATORS 348
FURTHER NOTES AND RESULTS 349
12 TOWARD A THEORY OF VECTOR-VALUED SPACES 352
12.1 SOME PROPERTIES OF ADMISSIBLE SPACES 352
12.2 SUBHARMONIC BEHAVIOR OF
||F(Z)||
X
359
12.2.1 BANACH ENVELOPE OF H
P
(X), 0 P 1 362
12.3 LINEAR OPERATORS ON HARDY AND BERGMAN SPACES 364
12.4 PROOF OF THE COIFMAN-ROCHBERG THEOREM 369
FURTHER NOTES AND RESULTS 374
A QUASI-BANACH SPACES 375
A.L QUASI-BANACH SPACES 375
A.2 Q-BANACH ENVELOPES 376
A.3 CLOSED GRAPH THEOREM 379
A.4 F-SPACES 382
A.4.1 NEVANLINNA CLASS 382
A.5 SPACES
P
383
A.6 LACUNARY SERIES IN QUASI-BANACH SPACES *
A.6.1 L
P
-INTEGRABILITY OF LACUNARY SERIES ON (0,1)
FURTHER NOTES AND RESULTS 395
B INTERPOLATION AND MAXIMAL FUNCTIONS 397
B.L RIESZ-THORIN THEOREM 397
B.2 WEAK L
P
-SPACES AND MARCINKIEWICZ'S THEOREM 399
B.3 CLASSICAL MAXIMAL FUNCTIONS 403
B.4 RADEMACHER FUNCTIONS AND KHINTCHIN'S INEQUALITY 409
* 384
385
CONTENTS
XIII
B.5 NIKISHIN'S THEOREM 410
B.6 NIKISHIN-STEIN'S THEOREM 412
B.7 BANACH'S PRINCIPLE AND THE THEOREM ON A.E. CONVERGENCE 415
B.8 VECTOR-VALUED MAXIMAL THEOREM 417
FURTHER NOTES AND RESULTS 418
BIBLIOGRAPHY 421
INDEX 443
The monograph contains a study on various function classes, a number of new results
and new or easy proofs of old results (Feff
er
man-Stein theorem on subharmonic behavior,
theorems on conjugate functions and fractional integration on Bergman spaces, Feffer-
man's duality theorem), which might be interesting for specialists; applications of the
Hardy-Littlewood inequalities on Taylor coefficients to
(C, öt)-maximal
theorems
and (C, r/)-convergence; a study of BMOA, due to Knese, based only on Green's
formula; the problem of membership of singular inner functions in
Besov
and
Hardy-Sobolev spaces; a full discussion on g-function (all
p
> 0)
and
Calderóni
area theorem; a new proof, due to
Astala
and Koskela, of the Littlewood-Paley
inequality for
univalent
functions; and new results and proofs on Lipschitz spaces,
coefficient multipliers and duality, including compact multipliers and multipliers on
spaces with non-normal weights. It also contains a discussion on analytic functions
and lacunary series with values in quasi-Banach spaces with applications to function
spaces and composition operators. Sixteen open questions are posed. |
| any_adam_object | 1 |
| author | Pavlović, Miroslav |
| author_GND | (DE-588)104661472X |
| author_facet | Pavlović, Miroslav |
| author_role | aut |
| author_sort | Pavlović, Miroslav |
| author_variant | m p mp |
| building | Verbundindex |
| bvnumber | BV041569791 |
| classification_rvk | SK 600 |
| ctrlnum | (OCoLC)864646496 (DE-599)DNB1032801077 |
| dewey-full | 510 515.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics 515 - Analysis |
| dewey-raw | 510 515.9 |
| dewey-search | 510 515.9 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041569791 |
| illustrated | Illustrated |
| indexdate | 2024-09-10T01:05:15Z |
| institution | BVB |
| isbn | 9783110281231 9783110281903 9783110281910 |
| language | English |
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| physical | XIII, 449 S. graph. Darst. 240 mm x 170 mm |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | De Gruyter |
| record_format | marc |
| series | De Gruyter Studies in Mathematics |
| series2 | De Gruyter Studies in Mathematics |
| spelling | Pavlović, Miroslav Verfasser (DE-588)104661472X aut Function classes on the unit disc an introduction Miroslav Pavlović 201312 Berlin De Gruyter 2014 XIII, 449 S. graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematics 52 Analytische Funktion (DE-588)4142348-3 gnd rswk-swf Komplexe Funktion (DE-588)4217733-9 gnd rswk-swf Komplexe Funktion (DE-588)4217733-9 s Analytische Funktion (DE-588)4142348-3 s DE-604 De Gruyter Studies in Mathematics 52 (DE-604)BV000005407 52 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4284947&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027015209&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027015209&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Pavlović, Miroslav Function classes on the unit disc an introduction De Gruyter Studies in Mathematics Analytische Funktion (DE-588)4142348-3 gnd Komplexe Funktion (DE-588)4217733-9 gnd |
| subject_GND | (DE-588)4142348-3 (DE-588)4217733-9 |
| title | Function classes on the unit disc an introduction |
| title_auth | Function classes on the unit disc an introduction |
| title_exact_search | Function classes on the unit disc an introduction |
| title_full | Function classes on the unit disc an introduction Miroslav Pavlović |
| title_fullStr | Function classes on the unit disc an introduction Miroslav Pavlović |
| title_full_unstemmed | Function classes on the unit disc an introduction Miroslav Pavlović |
| title_short | Function classes on the unit disc |
| title_sort | function classes on the unit disc an introduction |
| title_sub | an introduction |
| topic | Analytische Funktion (DE-588)4142348-3 gnd Komplexe Funktion (DE-588)4217733-9 gnd |
| topic_facet | Analytische Funktion Komplexe Funktion |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4284947&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027015209&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027015209&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
| work_keys_str_mv | AT pavlovicmiroslav functionclassesontheunitdiscanintroduction |