Bellman Function Technique in Harmonic Analysis

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.

'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

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 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).