Analysis in Banach spaces: Volume 1 Martingales and Littlewood-Paley theory
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2016]
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge, Volume 63 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvi, 614 Seiten |
| ISBN: | 9783319485195 |
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MARC
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Record in the Search Index
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| adam_text | Contents 1 Bochner spaces....................... ......................................................... . 1.1 Measurability................................................................... 1.1.a Functions on a measurable space (S,,e/)................. . 1.1.b Functions on a measure space (S,,í/, μ) ......................... 1.1.c Operator-valued functions................................................ 1.2 Integration ..................................................................................... 1.2.a The Bochner integral................. ...................................... 1.2.b The Bochner spaces Lv(5; X).......................................... 1.2.C The Pettis integral ............................................................ 1.3 Duality of Bochner spaces.......................... 1.3.a Elementary duality results....................... :....................... 1.3.b Duality and the Radon-Nikodým property.................... 1.3.c More about the Radon-Nikodým property .................. 1.4 Notes.................................. 1 2 2 8 12 13 13 21 30 36 37 40 48 59 2 Operators on Bochner spaces........................................................ 67 2.1 The Lp-extension problem.......................... 68 2.1.a Boundedness of T 8 Ιχ for positive operators T........... 70 2.1.b Boundedness of T ®1ң for Hilbert spaces II ............... 73 2.1.C Counterexamples............... 78 2.2 Interpolation of Bochner spaces.................................................. 83 2.2.a The Riesz-Thorin interpolation theorem....................... . 83 2.2.b The
Marcinkiewicz interpolation theorem....................... 86 2.2.С Complex interpolation of the spacesLP(S ;X)................. 91 2.2.d Real interpolation of the spaces LP(S:X)....................... 96 2.3 The Hardy-Littlewood maximal operator................................... 98 2.3.a Lebesgue points and differentiation ................................. 100 2.3.b Convolutions and approximation...................................... 103 2.4 The Fourier transform ................................................................... 105 2.4.a The inversion formula and Planchereľs theorem............106 2.4.b Fourier type ........................................................................110
x Contents 2.4.c The Schwartz class JX(Md՝,X)........................................... 116 2.4.d The space of tempered distributions 118 2.5 Sobolev spaces and differentiability..............................................122 2.5.a Weak derivatives............................................... 122 2.5.b The Sobolev spaces Wk*(D; X)....................... 123 2.5.C Almost everywheredifferentiability...................................124 2.5.d The fractional Sobolev spaces Ws’p(Kd; X) ....................133 2.6 Conditional expectations...............................................................137 2.6. a Uniqueness............................... 140 2.6.b Existence..................... ......................................................143 2.6.C Conditional limittheorems................................................. 146 2.6.d Inequalities and identities ........... 148 2.7 Notes......................... 154 3 Martingales.......................................................................................... 165 3.1 Definitions and basic properties.................................. ..166 3.1.a Difference sequences.......................................................... 167 3.1.b Paley-Walsh martingales...................................................168 3.1.c Stopped martingales.......................................................... 170 3.2 Martingale inequalities...................................................................173 3.2.a Doob’s maximal inequalities..............................................173 3.2.b Rademacher variables and
contractionprinciples ............180 3.2.C John-Nirenberg and Kahane-Khintchine inequalities ... 186 3.2.d Applications to inequalities on Rá.............. 193 3.3 Martingale convergence ............. 199 3.3.a Forward convergence..........................................................200 3.3.b Backward convergence......................................................201 3.3.0 The Itô-Nisio theorem for martingales...........................207 3.3.d Martingale convergence and the RNP............................. 211 3.4 Martingale decompositions.............................................................213 3.4.a Gundy decomposition........................................................ 214 3.4.b Davis decomposition................ 218 3.5 Martingale transforms............................................ 220 3.5.a Basic properties..................................................................220 3.5.b Extrapolation of Ap-inequalities................................. 229 3.5.C End-point estimates in A1................................................ 234 3.5.d Martingale type and cotype....................... 238 3.6 Approximate models for martingales............................................244 3.6.a Universality of Paley-Walsh martingales ....................... 244 3.6.b The Rademacher maximal function..................................249 3.6.С Approximate models for martingale transforms...............255 3.7 Notes................................................................................................259
Contents xi 4 UMD spaces..................... ..................................................................267 4.1 Motivation ...................................................................................... 268 4.1.a Square functions for martingale difference sequences ... 268 4.1.b Unconditionality.............................. 269 4.2 The UMD property.........................................................................281 4.2.a Definition and basic properties........................................ 281 4.2.b Unconditionality of the Haar decomposition ................. 286 4.2.C Examples and constructions............. .............................. 290 4.2.d Stein’s inequality for conditional expectations............... 298 4.2.e Boundedness of martingale transforms ...........................300 4.3 Banach space properties implied by UMD ..................................302 4.3.a Reflexivity................................................. 302 4.3.b Further Banach space properties implied by UMD......... 309 4.3.C Qiu’s example ............. 313 4.4 Decoupling and tangency .............................................................319 4.4.a Elementary decoupling...................................................... 319 4.4.b Tangent sequences..............................................................321 4.5 Burkholder functions and sharp UMD constants........................ 330 4.5.a Concave functions.................. 330 4.5.b Burkholder’s theorem........................................................ 332 4.5.c Optimal constants for the real
line................................... 337 4.5.d Differential subordination ................................................. 345 4.6 Notes................................................................................................354 5 Hilbert transform and Littlewood֊ Paley theory..................... 373 5.1 The Hilbert transform as a singular integral .............................. 374 5.1.a Dyadic shifts and their averages................. .....................375 5.1.b The Hilbert transform from the dyadicshifts................... 384 5.2 Fourier analysis of the Hilbert transform .................................... 387 5.2.a The Hilbert transform via the Fourier transform....... 388 5.2.b Periodic Hilbert transform and Fourierseries .................. 390 5.2.С Necessity of the UMD condition........................................395 5.3 Fourier multipliers...........................................................................399 5.3.a General theory...................................... 400 5.3.b iî-boundedness: a necessary condition for multipliers ... 407 5.3.C Mihlin’s multiplier theorem on К......................................409 5.3.d Littlewood-Paley inequalities on К..................................418 5.4 Applications to analysis in the Schatten classes.......................... 420 5.4.a The UMD property of the Schatten Classes ................... 420 5.4.b Schur multipliers and transference on гг?р....................... 423 5.4.c Operator Lipschitz functions ....................................... . 427 5.5 Fourier multipliers on
...............................................................430 5.5.a Riesz transforms and other liftings from R..................... 430 5.5.b Mihlin’s multiplier theorem on ....................................435 5.5.С Littlewood-Paley inequalities on ................................ 443
xii Contents 5.6 Applications to Sobolev spaces................................................... 448 5.6.a Bessel potential spaces......................................................... 448 5.6.b Complex interpolation of Bessel potential spaces ........... 452 5.6.C Coincidence of Sobolev and Bessel potential spaces .... 453 5.7 Transference and Fourier multipliers on Td ................................. 457 5.7.a Transference from Rd to T*................................................. 458 5.7.b Transference from Td to Rd................................................. 460 5.7.C Periodic multiplier theorems............................................... 463 5.8 Notes..................................................................................................469 О Open problems....................................................................................... 493 A Measure theory....................................................................................... 501 A.l Measure spaces.................................................................................. 501 A.l.a Basic definitions......................................... 501 A.l.b The structure of sub-σ-algebras......................................... 504 A.l.c Divisibility ............................................................................. 505 A.2 Convergence in measure.................................................................. 510 A.3 Uniform integrability ...................................................................... 512 A.4
Notes.................................................................................................. 515 В Banach spaces......................................................................................... 517 B.l Duality .............................................................................................. 517 B.l.a. Hahn- Banach theorems....................................................... 517 B.l.b Weak topologies..................................................................... 520 B.l.c Reflexivity............................................ 526 В.2 Bounded linear operators...................................... 526 B.3 Holomorphic mappings.................................................................... 529 B.4 Complexification .............................................................................. 530 B.5 Notes.................................................................................................. 533 C Interpolation theory............................................................................. 537 C.l Interpolation couples........................................................................537 C.2 Complex interpolation .................................................................... 538 C.3 Real interpolation............................................................................545 C.4 Complex versus real........................................ 559 C.5 Notes.................................................................................................. 562 D Schatten classes
..................................................................................... 565 D.l Approximation numbers and Schatten classes............................ 565 D.2 Holder’s inequality and duality......................................................570 D.3 Interpolation...................................................................................... 574 D.4 Notes..................................................................................................575 References........................................................................................................577 Index 605
|
| any_adam_object | 1 |
| author | Hytönen, Tuomas Neerven, Jan van 1964- Veraar, Mark 1980- |
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| author_facet | Hytönen, Tuomas Neerven, Jan van 1964- Veraar, Mark 1980- |
| author_role | aut aut aut |
| author_sort | Hytönen, Tuomas |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-07-10T07:41:09Z |
| institution | BVB |
| isbn | 9783319485195 |
| language | English |
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| spelling | Hytönen, Tuomas Verfasser (DE-588)1123529701 aut Analysis in Banach spaces Volume 1 Martingales and Littlewood-Paley theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis Cham Springer [2016] xvi, 614 Seiten txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Volume 63 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Banach-Raum (DE-588)4004402-6 gnd rswk-swf Littlewood-Paley-Theorem (DE-588)4352642-1 gnd rswk-swf Banach-Raum (DE-588)4004402-6 s Littlewood-Paley-Theorem (DE-588)4352642-1 s DE-604 Neerven, Jan van 1964- Verfasser (DE-588)1089478208 aut Veraar, Mark 1980- Verfasser (DE-588)1123530106 aut (DE-604)BV044011908 1 Erscheint auch als Online-Ausgabe 978-3-319-48520-1 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Volume 63 (DE-604)BV000899194 63 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge (DE-604)BV000899194 3 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=029419592&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hytönen, Tuomas Neerven, Jan van 1964- Veraar, Mark 1980- Analysis in Banach spaces Ergebnisse der Mathematik und ihrer Grenzgebiete Banach-Raum (DE-588)4004402-6 gnd Littlewood-Paley-Theorem (DE-588)4352642-1 gnd |
| subject_GND | (DE-588)4004402-6 (DE-588)4352642-1 |
| title | Analysis in Banach spaces |
| title_auth | Analysis in Banach spaces |
| title_exact_search | Analysis in Banach spaces |
| title_full | Analysis in Banach spaces Volume 1 Martingales and Littlewood-Paley theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis |
| title_fullStr | Analysis in Banach spaces Volume 1 Martingales and Littlewood-Paley theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis |
| title_full_unstemmed | Analysis in Banach spaces Volume 1 Martingales and Littlewood-Paley theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis |
| title_short | Analysis in Banach spaces |
| title_sort | analysis in banach spaces martingales and littlewood paley theory |
| topic | Banach-Raum (DE-588)4004402-6 gnd Littlewood-Paley-Theorem (DE-588)4352642-1 gnd |
| topic_facet | Banach-Raum Littlewood-Paley-Theorem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029419592&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV044011908 (DE-604)BV000899194 |
| work_keys_str_mv | AT hytonentuomas analysisinbanachspacesvolume1 AT neervenjanvan analysisinbanachspacesvolume1 AT veraarmark analysisinbanachspacesvolume1 |