Martingales in Banach Spaces

Description

This book focuses on the major applications of martingales to the geometry of Banach spaces, and a substantial discussion of harmonic analysis in Banach space valued Hardy spaces is also presented. It covers exciting links between super-reflexivity and some metric spaces related to computer science, as well as an outline of the recently developed theory of non-commutative martingales, which has natural connections with quantum physics and quantum information theory. Requiring few prerequisites and providing fully detailed proofs for the main results, this self-contained study is accessible to graduate students with a basic knowledge of real and complex analysis and functional analysis. Chapters can be read independently, with each building from the introductory notes, and the diversity of topics included also means this book can serve as the basis for a variety of graduate courses. Contents......Page 6 Introduction......Page 10 Description of the contents......Page 14 1.1 Banach space valued Lp-spaces......Page 30 1.2 Banach space valued conditional expectation......Page 36 1.3 Martingales: basic properties......Page 38 1.4 Examples of filtrations......Page 41 1.5 Stopping times......Page 46 1.6 Almost sure convergence: Maximal inequalities......Page 48 1.7 Independent increments......Page 57 1.8 Phillips’s theorem......Page 60 1.9 Reverse martingales......Page 63 1.10 Continuous time*......Page 65 1.11 Notes and remarks......Page 70 2.1 Vector measures......Page 71 2.2 Martingales, dentability and the Radon-Nikodým property......Page 75 2.3 The dual of Lp(B)......Page 86 2.4 Generalizations of Lp(B)......Page 89 2.5 The Krein-Milman property......Page 92 2.6 Edgar’s Choquet theorem......Page 96 2.7 The Lewis-Stegall theorem......Page 99 2.8 Notes and remarks......Page 102 3.1 Harmonicity and the Poisson kernel......Page 105 3.2 The hp spaces of harmonic functions on D......Page 109 3.3 Non-tangential maximal inequalities: boundary behaviour......Page 116 3.4 Harmonic functions and RNP......Page 126 3.5 Brownian martingales*......Page 130 3.6 Notes and remarks......Page 139 4.1 Subharmonic functions......Page 141 4.2 Outer functions and Hp(D)......Page 145 4.3 Banach space valued Hp-spaces for 0 < p ≤ ∞......Page 147 4.4 Analytic Radon-Nikodým property......Page 156 4.5 Hardy martingales and Brownian motion*......Page 160 4.6 B-valued hp and Hp over the half-plane U*......Page 169 4.7 Further complements*......Page 175 4.8 Notes and remarks......Page 177 5.1 Martingale transforms (scalar case): Burkholder’s inequalities......Page 180 5.2 Square functions for B-valued martingales: Kahane’s inequalities......Page 183 5.3 Definition of UMD......Page 188 5.4 Gundy’s decomposition......Page 191 5.5 Extrapolation......Page 197 5.6 The UMD1 property: Burgess Davis decomposition......Page 204 5.7 Examples: UMD implies super-RNP......Page 209 5.8 Dyadic UMD implies UMD......Page 212 5.9 The Burkholder-Rosenthal inequality......Page 219 5.10 Stein inequalities in UMD spaces......Page 224 5.11 Burkholder’s geometric characterization of UMD space......Page 226 5.12 Appendix: hypercontractivity on {−1, 1}......Page 235 5.13 Appendix: Hölder-Minkowski inequality......Page 236 5.14 Appendix: basic facts on weak-Lp......Page 238 5.15 Appendix: reverse Hölder principle......Page 239 5.16 Appendix: Marcinkiewicz theorem......Page 241 5.18 Notes and remarks......Page 243 6.1 Hilbert transform: HT spaces......Page 247 6.2 Bourgain’s transference theorem: HT implies UMD......Page 257 6.3 UMD implies HT......Page 262 6.4 UMD implies HT (with stochastic integrals)*......Page 275 6.5 Littlewood-Paley inequalities in UMD spaces......Page 278 6.6 The Walsh system Hilbert transform......Page 283 6.7 Analytic UMD property*......Page 284 6.8 UMD operators*......Page 286 6.9 Notes and remarks......Page 289 7.1 Banach space valued H1 and BMO: Fefferman’s duality theorem......Page 292 7.2 Atomic B-valued H1......Page 295 7.3 H1, BMO and atoms for martingales......Page 311 7.4 Regular filtrations......Page 320 7.5 From dyadic BMO to classical BMO......Page 321 7.6 Notes and remarks......Page 326 8 Interpolation methods (complex and real)......Page 328 8.1 The unit strip......Page 329 8.2 The complex interpolation method......Page 332 8.3 Duality for the complex method......Page 350 8.4 The real interpolation method......Page 361 8.5 Real interpolation between Lp-spaces......Page 365 8.6 The K-functional for (L1(B0 ), L∞(B1 ))......Page 371 8.7 Real interpolation between vector valued Lp-spaces......Page 375 8.8 Duality for the real method......Page 381 8.9 Reiteration for the real method......Page 386 8.10 Comparing the real and complex methods......Page 391 8.11 Symmetric and self-dual interpolation pairs......Page 392 8.12 Notes and remarks......Page 397 9 The strong p-variation of scalar valued martingales......Page 401 9.1 Notes and remarks......Page 409 10.1 Uniform convexity......Page 411 10.2 Uniform smoothness......Page 424 10.3 Uniform convexity and smoothness of Lp......Page 433 10.4 Type, cotype and UMD......Page 435 10.5 Square function inequalities in q-uniformly convex and p-uniformly smooth spaces......Page 446 10.6 Strong p-variation, uniform convexity and smoothness......Page 452 10.7 Notes and remarks......Page 454 11.1 Finite representability and super-properties......Page 457 11.2 Super-reflexivity and inequalities for basic sequences......Page 462 11.3 Uniformly non-square and J-convex spaces......Page 475 11.4 Super-reflexivity and uniform convexity......Page 483 11.5 Strong law of large numbers and super-reflexivity......Page 486 11.6 Complex interpolation: θ-Hilbertian spaces......Page 488 11.7 Complex analogues of uniform convexity*......Page 492 11.8 Appendix: ultrafilters, ultraproducts......Page 501 11.9 Notes and remarks......Page 503 12 Interpolation between strong p-variation spaces......Page 506 12.1 The spaces vp(B), Wp(B) and Wp,q(B)......Page 507 12.2 Duality and quasi-reflexivity......Page 511 12.3 The intermediate spaces up(B) and vp(B)......Page 514 12.4 Lq-spaces with values in vp and Wp......Page 518 12.5 Some applications......Page 521 12.6 K-functional for (vr (B), ℓ∞(B))......Page 523 12.7 Strong p-variation in approximation theory......Page 525 12.8 Notes and remarks......Page 528 13.1 Exponential inequalities......Page 529 13.2 Concentration of measure......Page 532 13.3 Metric characterization of super-reflexivity: trees......Page 535 13.4 Another metric characterization of super-reflexivity: diamonds......Page 541 13.5 Markov type p and uniform smoothness......Page 548 13.6 Notes and remarks......Page 550 14.1 Non-commutative probability space......Page 552 14.2 Non-commutative Lp-spaces......Page 553 14.3 Conditional expectations: non-commutative martingales......Page 556 14.4 Examples......Page 558 14.5 Non-commutative Khintchin inequalities......Page 560 14.6 Non-commutative Burkholder inequalities......Page 562 14.7 Non-commutative martingale transforms......Page 564 14.8 Non-commutative maximal inequalities......Page 566 14.9 Martingales in operator spaces......Page 568 14.10 Notes and remarks......Page 570 Bibliography......Page 571 Index......Page 589