Analysis in Banach spaces. Volume I, Martingales and Littlewood-Paley theory /
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| Author / Creator: | Hytönen, Tuomas, author. |
|---|---|
| Imprint: | Cham : Springer, 2016. |
| Description: | 1 online resource (628 pages) |
| Language: | English |
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ; v. 63 Ergebnisse der Mathematik und ihrer Grenzgebiete ; v. 63. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11269719 |
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| 100 | 1 | |a Hytönen, Tuomas, |e author. |0 http://id.loc.gov/authorities/names/nb2017001644 | |
| 245 | 1 | 0 | |a Analysis in Banach spaces. |n Volume I, |p Martingales and Littlewood-Paley theory / |c Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis, authors. |
| 246 | 3 | 0 | |a Martingales and Littlewood-Paley theory |
| 264 | 1 | |a Cham : |b Springer, |c 2016. | |
| 300 | |a 1 online resource (628 pages) | ||
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| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ; |v v. 63 | |
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Preface; Contents; Symbols and notations; Standing assumptions; 1 Bochner spaces; 1.1 Measurability; 1.1.a Functions on a measurable space (S; A); 1.1.b Functions on a measure space (S; A ;); 1.1.c Operator-valued functions; 1.2 Integration; 1.2.a The Bochner integral; 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; 2 Operators on Bochner spaces; 2.1 The Lp-extension problem ; 2.1.a Boundedness of T IX for positive operators T. | |
| 505 | 8 | |a 2.1.b Boundedness of T IH for Hilbert spaces H2.1.c Counterexamples; 2.2 Interpolation of Bochner spaces; 2.2.a The Riesz-Thorin interpolation theorem; 2.2.b The Marcinkiewicz interpolation theorem; 2.2.c Complex interpolation of the spaces Lp(S; X); 2.2.d Real interpolation of the spaces Lp(S; X); 2.3 The Hardy-Littlewood maximal operator; 2.3.a Lebesgue points and differentiation; 2.3.b Convolutions and approximation; 2.4 The Fourier transform; 2.4.a The inversion formula and Plancherel's theorem; 2.4.b Fourier type; 2.4.c The Schwartz class S(Rd; X). | |
| 505 | 8 | |a 2.4.d The space of tempered distributions S0(RdX); 2.5 Sobolev spaces and differentiability; 2.5.a Weak derivatives; 2.5.b The Sobolev spaces Wk; p(D; X); 2.5.c Almost everywhere differentiability; 2.5.d The fractional Sobolev spaces Ws; p(Rd; X); 2.6 Conditional expectations; 2.6.a Uniqueness; 2.6.b Existence; 2.6.c Conditional limit theorems; 2.6.d Inequalities and identities; 2.7 Notes; 3 Martingales; 3.1 Definitions and basic properties; 3.1.a Difference sequences; 3.1.b Paley-Walsh martingales; 3.1.c Stopped martingales; 3.2 Martingale inequalities; 3.2.a Doob's maximal inequalities. | |
| 505 | 8 | |a 3.2.b Rademacher variables and contraction principles3.2.c John-Nirenberg and Kahane-Khintchine inequalities; 3.2.d Applications to inequalities on; 3.3 Martingale convergence; 3.3.a Forward convergence; 3.3.b Backward convergence; 3.3.c The Itô-Nisio theorem for martingales; 3.3.d Martingale convergence and the RNP; 3.4 Martingale decompositions; 3.4.a Gundy decomposition; 3.4.b Davis decomposition; 3.5 Martingale transforms; 3.5.a Basic properties; 3.5.b Extrapolation of Lp-inequalities; 3.5.c End-point estimates in L1 ; 3.5.d Martingale type and cotype. | |
| 505 | 8 | |a 3.6 Approximate models for martingales3.6.a Universality of Paley-Walsh martingales; 3.6.b The Rademacher maximal function; 3.6.c Approximate models for martingale transforms; 3.7 Notes; 4 UMD spaces; 4.1 Motivation; 4.1.a Square functions for martingale difference sequences; 4.1.b Unconditionality; 4.2 The UMD property; 4.2.a Definition and basic properties; 4.2.b Unconditionality of the Haar decomposition; 4.2.c Examples and constructions; 4.2.d Stein's inequality for conditional expectations; 4.2.e Boundedness of martingale transforms; 4.3 Banach space properties implied by UMD. | |
| 500 | |a 4.3.a Reflexivity. | ||
| 504 | |a Includes bibliographical references and index. | ||
| 520 | |a The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem. Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas. | ||
| 650 | 0 | |a Banach spaces. |0 http://id.loc.gov/authorities/subjects/sh85011441 | |
| 650 | 0 | |a Littlewood-Paley theory. |0 http://id.loc.gov/authorities/subjects/sh94002140 | |
| 650 | 0 | |a Martingales (Mathematics) |0 http://id.loc.gov/authorities/subjects/sh85081645 | |
| 650 | 7 | |a MATHEMATICS |x Calculus. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Mathematical Analysis. |2 bisacsh | |
| 650 | 7 | |a Banach spaces. |2 fast |0 (OCoLC)fst00826389 | |
| 650 | 7 | |a Littlewood-Paley theory. |2 fast |0 (OCoLC)fst01000542 | |
| 650 | 7 | |a Martingales (Mathematics) |2 fast |0 (OCoLC)fst01010880 | |
| 655 | 0 | |a Electronic books. | |
| 655 | 4 | |a Electronic books. | |
| 700 | 1 | |a Neerven, Jan van, |d 1964- |e author. |0 http://id.loc.gov/authorities/names/n92100276 | |
| 700 | 1 | |a Veraar, Mark, |e author. |0 http://id.loc.gov/authorities/names/nb2017001650 | |
| 700 | 1 | |a Weis, Lutz, |e author. |0 http://id.loc.gov/authorities/names/n00007142 | |
| 776 | 0 | 8 | |i Print version: |a Hytönen, Tuomas. |t Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory. |d Cham : Springer International Publishing, ©2016 |z 9783319485195 |
| 830 | 0 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete ; |v v. 63. | |
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