**RESEARCH IN PAIRS**

**Bellman Functions Method in Harmonic Analysis**

*Fonctions de Bellman en analyse harmonique***5-16 March 2018**

Donald Burkholder invented what is now known as “the method of Bellman function” for the goal of the solution of Pelczynski’s problem of finding the sharp constants of unconditional basis consisting of Haar functions. Nazarov, Treil and Volberg at the mid-90’s observed that the method of Burkholder is indeed the application of the value function of stochastic optimal control. This direct analogy, understood correctly, allowed them to apply this method to a wide class of harmonic analysis problem, in fact, to all problems that have a certain intrinsic Markov property. Recently, some applications to Big Data theory also appeared in the works of P. Ivanisvili and A. Volberg. Already, many previously unsolved problems of harmonic analysis were solved in this way. To name just a few: 1) complete characterization of the two weight problem for the martingale transform, 2) the first results towards A2 conjecture (together with S. Petermichl), 3) the solution (in the negative) of Sarason’s two weight conjecture and Muckenhoupt–Wheeden’s A1 conjectures, 4) the solution of Astala–Iwaniec–Saksman problem concerning the regularity of the solutions of Beltrami equations for the end-point exponent (with S. Petermichl). The method continues its development, in particular, harmonic analysis Bellman functions of 2 variables got thorough description in the works of Vasyunin and his collaborators. This allows them to write completely sharp estimates in various type of John–Nirenberg inequalities and related topics of BMO and weighted theory. The time has come to finish the book that I, Alexander Volberg, am writing jointly with Vasily Vasyunin. It has 300 pages, and we have to meet to produce a final version together.