The Bellman Function Technique in Harmonic Analysis

A comprehensive reference on the Bellman function method and its applications to various topics in probability and harmonic analysis.

Vasily Vasyunin (Author), Alexander Volberg (Author)

9781108486897, Cambridge University Press

Hardback, published 6 August 2020

460 pages
23.4 x 15.6 x 3.1 cm, 0.77 kg

'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

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.

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.

Subject Areas: Probability & statistics [PBT], Calculus & mathematical analysis [PBK]