Analysis in Banach spaces: Volume 2 Probabilistic methods and operator theory
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2017]
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge, volume 67 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxi, 616 Seiten |
| ISBN: | 9783319698076 |
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MARC
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Record in the Search Index
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| adam_text | Contents 6 Random sums............................................. ............................................. 6.1 Basic notions and estimates................................................................ 6.1.a Symmetric random variables and randomisation............. 6.1.b Kahane’s contraction principle............................................. 6.1.c Norm comparison of different random sums..................... 6.1.d Covariance domination for Gaussian sums........................ 6.2 Comparison of different Lp-norms .................................................... 6.2.a The discrete heat semigroup and hypercontractivity ... 6.2.b Kahane-Khiņtchine inequalities........................................... 6.2.C End-point bounds related to p = 0 and q .= oo . ............. 6.3 The random sequence spaces ερ(Χ) and yp(X)........ ................... 6.3.a Coincidence with square function spaces when X — Lq . 6.3.b Dual and bi-dual of ερΝ{Χ) and 7^(X).............................. 6.4 Convergence of random series............................................................ 6.4.a Itô-Nisio equivalence of different modes of convergence . 6.4.b Boundedness implies convergence if and only if Co $7 X . 6.5 Comparison of random sums and trigonometric sums................. 6.6 Notes.......................................................................... 1 2 4 9 10 15 17 18 21 24 26 28 29 33 33 39 42 46 7 Type, cotype, and related properties............................................... 7.1 Type and
cotype................................................................................ 7.1.a Definitions and basic properties........................................... 7.1.b Basic examples.................................. ........................................ 7.1.C Type implies cotype................................................................ 7.1.d Type and cotype for general random sequences.............. 7.1.e Extremality of Gaussians in (co)type 2 spaces ............... 7.2 Comparison theorems under finite cotype.................................... 7.2.a Summing operators.................................................................. 7.2.b Pisier’s factorisation theorem............................................... 7.2.c Contraction principle with function coefficients ............. 7.2.d Equivalence of cotype and Gaussian cotype.................... 53 54 54 56 63 66 69 73 74 75 79 81
xii Contents 7.2.e Finite cotype in Banach lattices...................................... 85 7.3 Geometric characterisations...................................... 88 7.3. a Kwapień’s characterisation of type and cotype 2........... 89 7.3.b Maurey-Pisier characterisation of non-trivial (co)type .. 96 7.4 if-convexity.................................................................................... 109 7.4.a Definition and basic properties................................ . 109 7.4.b if-convexity and type......................................................... 114 7.4.c if-convexity and duality of the spaces ερΝ{Χ)................115 7.4.d if-convexity and interpolation .........................................118 7.4.e if-convexity with respect to general random variables .. 120 7.4.f Equivalence of if-convexity and non-trivial type....... 123 7.5 Contraction principles for double random sums.......................... 129 7.5.a Pisier’s contraction property............................................ 130 7.5.b The triangular contraction property ............. ................136 7.5.C Duality and interpolation................................................... 140 7.5.d Gaussian version of Pisier’s contraction property........ 142 7.5.e Double random sums in Banach lattices......................... 145 7.6 Notes...................................... 146 8 ß-boundedness................................................. 163 8.1 Basic theory.......................................... 164 8.1.a Definition and comparison with related notions............. 164 8.1.b Testing
R-boundedness with distinct operators............. 168 8.1.C First examples: multiplication and averaging operators . 170 8.1.d R-boundedness versus boundedness on 4 PO.................177 8.1.e Stability of R-boundedness under set operations.............178 8.2 Sources of R-boundedness in real analysis ..................................183 8.2.a Pointwise domination by the maximal operator............. 183 8.2.b Inequalities with Muckenhoupt weights............................ 185 8.2.C Characterisation by weighted inequalities in Lp...............187 8.3 Fourier multipliers and R-boundedness........................................192 8.3.a Multipliers of bounded variation on the line................... 192 8.3.b The Marcinkiewicz multiplier theorem on the line ........197 8.3.C Multipliers of bounded rectangular variation ..................200 8.3.d The product-space multiplier theorem.............................205 8.3.e Necessity of Pisier’s contraction property....................... 209 8.4 Sources of R-boundedness in operator theory.............................. 211 8.4.a Duality and interpolation................................................... 211 8.4.b Unconditionality.................................................................214 8.5 Integral means and smooth functions........................................... 217 8.5.a Integral means I: elementary estimates...........................217 8.5.b The range of differentiable and holomorphic functions .. 221 8.5.C Integral means II: the effect of type and cotype .............225 8.5.d The range of functions
of fractional smoothness.............231 8.6 Coincidence of R-boundedness with other notions...................... 234
Contents 8.7 9 xiii 8.6.a Coincidence with boundedness implies (co)type 2............ 234 8.6.b Coincidence with 7-boundedness implies finite cotype .. 237 Notes................... 241 Square functions and radonifying operators................................251 9.1 Radonifying operators........................................................................ 252 9.1.a Heuristics........................................................ 252 9.1.b The operator spaces7^(Η,Χ) and 7(H,X).....................254 9.1.C The ideal property ................................................................... 260 9.1.d Convergence results............................... 261 9.1.e Coincidence 700 (H,X) = 7(H,X) when Co % X.............. 265 9.1.f Trace duality.............................................................................267 9.1.g Interpolation...................................... 269 9.1.Һ The indefinite integral and Brownian motion................. 271 9.2 Functions representing a 7-radonifying operator........................... 275 9.2.a Definitions and basic properties...........................................275 9.2.b Square integrability versus 7-radonification.......................278 9.2.C Trace duality and the 7-Hölder inequality........ ................282 9.3 Square function characterisations.......................................................284 9.3.a Square functions in Մ-spaces...............................................284 9.3.b Square functions in Banach function spaces.......................287 9.4 Function space
properties............................................ 293 9.4.a Convergence theorems ............................. ........................... .293 9.4.b Fubini-type theorems............................................................... 296 9.4.C . Partitions, type and cotype................................................... 298 9.5 The 7-multiplier theorem......................................................................299 9.5.a Sufficient conditions for pointwise multiplication............. 300 9.5.b Necessity of 7-boundedness....................... 303 9.6 Extension theorems............................... 308 9.6.a General extension results.........................................................308 9.6.b Л-bounded extensions via Pisier’s contraction property . 311 9.6.C Л-bounded extensions via type and cotype........ ................314 9.7 Function space embeddings . ................................................................320 9.7.a Embedding Sobolev spaces ................................................... 321 9.7.b Embedding Hôlder spaces.......................................................326 9.7.C Embeddings involving holomorphic functions................... 329 9.7.d Hilbert sequences........................................................................337 9.8 Notes..................... 345 10 The H°°-functional calculus ........................... .................................... 359 10.1 Sectorial operators...................................................................................360 10.1.a Examples
.....................................................................................362 10.1.b Basic properties.......................................................................... 364 10.2 Construction of the if°°-calculus............................... 368 10.2.a The Dunford calculus ............................................... .............. 369
XIV Contents 10.3 10.4 10.5 10.6 10.7 10.8 10.2.b The iř°°-calculus.................................................................379 10.2.c First examples.................................................................... 388 Д-boundedness of the H°°-calculus............................................. 398 10.3.a Я-sectoriality...................................................................... 399 10.3.b The main Д-boundedness result .......................................401 10.3.C Unconditional decompositions associated with A ......... 404 10.3.d Proof of the main result....................................................406 Square functions and íř°°-calculus..............................................413 10.4.a Discrete square functions..................................................414 10.4.b Continuous square functions............................................. 422 10.4.c The quadratic H°°-calculus...................................... 432 10.4.d Le Merdy’s theorem on the angle of the iř°°-calculus .. 435 Necessity of UMD and Pisier’s contraction property.................. 440 The bisectorial iT^-calculus.........................................................446 10.6.a Bisectorial operators...........................................................446 10.6.b Basic theory of the bisectorial calculus............................ 449 Functional calculus for (semi)group generators.......................... 451 10.7.a The Phillips calculus...........................................................452 10.7.b Coifman-Weiss transference
theorems..............................456 10.7.C Hieber-Priiss theorems on U°°-calculus for generators . 461 10.7.d Analytic semigroups of positive contractions on Lp .... 462 Notes......... ..................................................................................... 468 P Problems............................................................................................... 479 E Probability theory.............................................................................. 487 E.l Random variables .......................................................................... 487 E.l.a Modes of convergence.........................................................488 E.l.b Independence...................................................................... 491 E.l.c Characteristic functions....................... 493 E.2 Gaussian variables.......................................................................... 495 E.2.a Multivariate Gaussian variables........................................498 E.2.b The central limit theorem ................................................. 502 E.2.C The maximum of Gaussian variables................................505 E.3 Notes................................................................................................509 F Banach lattices ......................................................... 511 F.l Definitions and basic properties...................... 511 F.2 The Krivine calculus...................................................................... 512 F.3 Lorentz
spaces................................................................................ 514 F.4 Notes................................................................................................517
Contents XV G Semigroups of linear operators....................................................... 519 G.l Unbounded linear operators........................................................... 519 G.2 Co-semigroups........................... .............................................. · · 523 G.3 The inhomogeneous abstract Cauchy problem...........................531 G.4 The Hille-Yosida theorem....................... 532 G.5 Analytic semigroups ............................................................. 536 G.6 Stone’s theorem................................. ■···.....................................543 G.7 Notes................................................ 545 H Hardy spaces of holomorphic functions ..................................... 549 H.l Hardy spaces on a strip............ ................................................. 549 H.2 Hardy spaces on a sector.......................................................... 552 H.3 The Franks-Mclntosh decomposition...........................................553 H.4 Notes............................................................................................... 559 I Akcoglu’s theory of positive contractions on Lv......................561 I.1 Isometric dilations..........................................................................561 1.2 Maximal ergodic averages ........... 569 1.3 Notes................................................................................................574 J Muckenhoupt weights ......................................................................577 J.l Weighted boundedness of
themaximal operator ................. 577 J.2 Rubio de Francia’s weightedextrapolation theorem................... 579 J.3 Notes.......................................................... 583 References........................................................................... ........................ 585 Index .610
|
| any_adam_object | 1 |
| author | Hytönen, Tuomas Neerven, Jan van 1964- Veraar, Mark 1980- |
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| dewey-full | 515.2433 |
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| dewey-ones | 515 - Analysis |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-07-10T08:03:17Z |
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| isbn | 9783319698076 |
| language | English |
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| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| spelling | Hytönen, Tuomas Verfasser (DE-588)1123529701 aut Analysis in Banach spaces Volume 2 Probabilistic methods and operator theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis Cham Springer [2017] xxi, 616 Seiten txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, volume 67 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Littlewood-Paley-Theorem (DE-588)4352642-1 gnd rswk-swf Banach-Raum (DE-588)4004402-6 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 2 Erscheint auch als Online-Ausgabe 978-3-319-69808-3 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, volume 67 (DE-604)BV000899194 67 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=030261488&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 Littlewood-Paley-Theorem (DE-588)4352642-1 gnd Banach-Raum (DE-588)4004402-6 gnd |
| subject_GND | (DE-588)4352642-1 (DE-588)4004402-6 |
| 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 2 Probabilistic methods and operator theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis |
| title_fullStr | Analysis in Banach spaces Volume 2 Probabilistic methods and operator theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis |
| title_full_unstemmed | Analysis in Banach spaces Volume 2 Probabilistic methods and operator theory Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis |
| title_short | Analysis in Banach spaces |
| title_sort | analysis in banach spaces probabilistic methods and operator theory |
| topic | Littlewood-Paley-Theorem (DE-588)4352642-1 gnd Banach-Raum (DE-588)4004402-6 gnd |
| topic_facet | Littlewood-Paley-Theorem Banach-Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030261488&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV044011908 (DE-604)BV000899194 |
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