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).

'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

• 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.