UID:
almafu_9960117310402883
Umfang:
1 online resource (xxviii, 561 pages) :
,
digital, PDF file(s).
Ausgabe:
First edition.
ISBN:
1-316-67895-4
,
1-316-68014-2
,
1-316-48058-5
Serie:
Cambridge Studies in Advanced Mathematics ; Volume 155
Inhalt:
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.
Anmerkung:
Title from publisher's bibliographic system (viewed on 06 Jun 2016).
,
Cover -- Half title -- Series -- Title -- Copyright -- Contents -- Introduction -- Description of the contents -- 1 Banach space valued martingales -- 1.1 Banach space valued Lp-spaces -- 1.2 Banach space valued conditional expectation -- 1.3 Martingales: basic properties -- 1.4 Examples of filtrations -- 1.5 Stopping times -- 1.6 Almost sure convergence: Maximal inequalities -- 1.7 Independent increments -- 1.8 Phillips's theorem -- 1.9 Reverse martingales -- 1.10 Continuous time* -- 1.11 Notes and remarks -- 2 Radon-Nikodým property -- 2.1 Vector measures -- 2.2 Martingales, dentability and the Radon-Nikodým property -- 2.3 The dual of Lp(B) -- 2.4 Generalizations of Lp(B) -- 2.5 The Krein-Milman property -- 2.6 Edgar's Choquet theorem -- 2.7 The Lewis-Stegall theorem -- 2.8 Notes and remarks -- 3 Harmonic functions and RNP -- 3.1 Harmonicity and the Poisson kernel -- 3.2 The h[sup(p)] spaces of harmonic functions on D -- 3.3 Non-tangential maximal inequalities: boundary behaviour -- 3.4 Harmonic functions and RNP -- 3.5 Brownian martingales* -- 3.6 Notes and remarks -- 4 Analytic functions and ARNP -- 4.1 Subharmonic functions -- 4.2 Outer functions and H[sup(p)](D) -- 4.3 Banach space valued H[sup(p)]-spaces for 0 < -- p ≤ ∞ -- 4.4 Analytic Radon-Nikodým property -- 4.5 Hardy martingales and Brownian motion* -- 4.6 B-valued h[sup(p)] and H[sup(p)] over the half-plane U* -- 4.7 Further complements* -- 4.8 Notes and remarks -- 5 The UMD property for Banach spaces -- 5.1 Martingale transforms (scalar case): Burkholder's inequalities -- 5.2 Square functions for B-valued martingales: Kahane's inequalities -- 5.3 Definition of UMD -- 5.4 Gundy's decomposition -- 5.5 Extrapolation -- 5.6 The UMD[sub(1)] property: Burgess Davis decomposition -- 5.7 Examples: UMD implies super-RNP -- 5.8 Dyadic UMD implies UMD.
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5.9 The Burkholder-Rosenthal inequality -- 5.10 Stein inequalities in UMD spaces -- 5.11 Burkholder's geometric characterization of UMD space -- 5.12 Appendix: hypercontractivity on {-1,1} -- 5.13 Appendix: Hölder-Minkowski inequality -- 5.14 Appendix: basic facts on weak-L[sub(p)] -- 5.15 Appendix: reverse Hölder principle -- 5.16 Appendix: Marcinkiewicz theorem -- 5.17 Appendix: exponential inequalities and growth of L[sub(p)]-norms -- 5.18 Notes and remarks -- 6 The Hilbert transform and UMD Banach spaces -- 6.1 Hilbert transform: HT spaces -- 6.2 Bourgain's transference theorem: HT implies UMD -- 6.3 UMD implies HT -- 6.4 UMD implies HT (with stochastic integrals)* -- 6.5 Littlewood-Paley inequalities in UMD spaces -- 6.6 The Walsh system Hilbert transform -- 6.7 Analytic UMD property* -- 6.8 UMD operators* -- 6.9 Notes and remarks -- 7 Banach space valued H[sup(1)] and BMO -- 7.1 Banach space valued H[sup(1)] and BMO:Fefferman's duality theorem -- 7.2 Atomic B-valued H[sup(1)] -- 7.3 H[sup(1)], BMO and atoms for martingales -- 7.4 Regular filtrations -- 7.5 From dyadic BMO to classical BMO -- 7.6 Notes and remarks -- 8 Interpolation methods (complex and real) -- 8.1 The unit strip -- 8.2 The complex interpolation method -- 8.3 Duality for the complex method -- 8.4 The real interpolation method -- 8.5 Real interpolation between Lp-spaces -- 8.6 The K-functional for (L[sub(1)](B[sub(0)] ), L[sub(∞)](B[sub(1)])) -- 8.7 Real interpolation between vector valued Lp-spaces -- 8.8 Duality for the real method -- 8.9 Reiteration for the real method -- 8.10 Comparing the real and complex methods -- 8.11 Symmetric and self-dual interpolation pairs -- 8.12 Notes and remarks -- 9 The strong p-variation of scalar valued martingales -- 9.1 Notes and remarks -- 10 Uniformly convex Banach space valued martingales -- 10.1 Uniform convexity.
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10.2 Uniform smoothness -- 10.3 Uniform convexity and smoothness of Lp -- 10.4 Type, cotype and UMD -- 10.5 Square function inequalities in q-uniformly convex and p-uniformly smooth spaces -- 10.6 Strong p-variation, uniform convexity and smoothness -- 10.7 Notes and remarks -- 11 Super-reflexivity -- 11.1 Finite representability and super-properties -- 11.2 Super-reflexivity and inequalities for basic sequences -- 11.3 Uniformly non-square and J-convex spaces -- 11.4 Super-reflexivity and uniform convexity -- 11.5 Strong law of large numbers and super-reflexivity -- 11.6 Complex interpolation: θ-Hilbertian spaces -- 11.7 Complex analogues of uniform convexity* -- 11.8 Appendix: ultrafilters, ultraproducts -- 11.9 Notes and remarks -- 12 Interpolation between strong p-variation spaces -- 12.1 The spaces v[sub(p)](B), W[sub(p)](B) and W[sub(p,q)](B) -- 12.2 Duality and quasi-reflexivity -- 12.3 The intermediate spaces u[sub(p)](B) and v[sub(p)](B) -- 12.4 L[sub(q)]-spaces with values in v[sub(p)] and W[sub(p)] -- 12.5 Some applications -- 12.6 K-functional for (v[sub(r)] (B), l[sub(∞)](B)) -- 12.7 Strong p-variation in approximation theory -- 12.8 Notes and remarks -- 13 Martingales and metric spaces -- 13.1 Exponential inequalities -- 13.2 Concentration of measure -- 13.3 Metric characterization of super-reflexivity: trees -- 13.4 Another metric characterization of super-reflexivity: diamonds -- 13.5 Markov type p and uniform smoothness -- 13.6 Notes and remarks -- 14 An invitation to martingales in non-commutative Lp-spaces* -- 14.1 Non-commutative probability space -- 14.2 Non-commutative Lp-spaces -- 14.3 Conditional expectations: non-commutative martingales -- 14.4 Examples -- 14.5 Non-commutative Khintchin inequalities -- 14.6 Non-commutative Burkholder inequalities -- 14.7 Non-commutative martingale transforms.
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14.8 Non-commutative maximal inequalities -- 14.9 Martingales in operator spaces -- 14.10 Notes and remarks -- Bibliography -- Index.
Weitere Ausg.:
ISBN 1-107-13724-1
Sprache:
Englisch
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