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The Bellman Function Technique in Harmonic Analysis

The Bellman Function Technique in Harmonic Analysis

£61.99

Part of Cambridge Studies in Advanced Mathematics

  • Publication planned for: May 2020
  • availability: Not yet published - available from May 2020
  • format: Hardback
  • isbn: 9781108486897

£ 61.99
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  • The Bellman function, a powerful tool originating in control theory, can be used successfully in a large class of difficult harmonic analysis problems and has produced some notable results over the last thirty years. This book by two leading experts is the first devoted to the Bellman function method and its applications to various topics in probability and harmonic analysis. Beginning with basic concepts, the theory is introduced step-by-step starting with many examples of gradually increasing sophistication, culminating with Calderón–Zygmund operators and end-point estimates. All necessary techniques are explained in generality, making this book accessible to readers without specialized training in non-linear PDEs or stochastic optimal control. Graduate students and researchers in harmonic analysis, PDEs, functional analysis, and probability will find this to be an incisive reference, and can use it as the basis of a graduate course.

    • Enables researchers in harmonic analysis to use Bellman function techniques in their own work
    • Introduces theories step-by-step using classical examples, including the work of Burkholder, as well as recent solutions of several outstanding problems
    • Demonstrates interdisciplinary interactions between stochastic control, non-linear PDE, and harmonic analysis
    • Offers a geometric perspective, translating linear infinite dimensional problems in functional analysis to finite dimensional non-linear problems
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    Reviews & endorsements

    'I first encountered Bellman functions about 35 years ago when advising engineers striving to minimize the expenditure of diamond chips in silicon grinding. Fifteen years later I was amused to learn that Nazarov, Treil, and Volberg successfully applied similar ideas to a variety of problems in harmonic analysis. Together with Vasyunin (and other analysts), they developed these techniques into a powerful tool which is carefully explained in the present book. The book is written on a level accessible to graduate students and I recommend it to everyone who wishes to join the Bellman functions club.' Mikhail Sodin, Tel Aviv University

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    Product details

    • Publication planned for: May 2020
    • format: Hardback
    • isbn: 9781108486897
    • dimensions: 228 x 152 mm
    • availability: Not yet published - available from May 2020
  • Table of Contents

    Introduction
    1. Examples of Bellman functions
    2. What you always wanted to know about Stochastic Optimal Control, but were afraid to ask
    3. Conformal martingales models. Stochastic and classical Ahlfors-Beurling operators
    4. Dyadic models. Application of Bellman technique to upper estimates of singular integrals
    5. Application of Bellman technique to the end-point estimates of singular integrals.

  • Authors

    Vasily Vasyunin, Russian Academy of Sciences
    Vasily Vasyunin is a Leading Researcher at the St Petersburg Department of the Steklov Mathematical Institute of Russian Academy of Sciences and Professor of Saint-Petersburg State University. His research interests include linear and complex analysis, operator models, and harmonic analysis. Vasyunin has taught at universities in Europe, and the United States. He has authored or co-authored over sixty articles.

    Alexander Volberg, Michigan State University
    Alexander L. Volberg is a Distinguished Professor of Mathematics at Michigan State University. He was the recipient of the Onsager Medal as well as the Salem Prize, awarded to a young researcher in the field of analysis. Along with teaching at institutions in Paris and Edinburgh, Volberg also served as a Humboldt senior researcher, Clay senior researcher, and a Simons fellow. He has co-authored 179 papers, and is the author of Calderon-Zygmund Capacities and Operators on Non-Homogenous Spaces (2004).

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