From harmonic analysis problems to Hamilton--Jacobi--Bellman PDE and back to harmonic analysis problems.

We will explain the Bellman function approach to some singular integral estimates. There is a dictionary that translates the language of singular integrals to the language of stochastic optimization. The main tool in stochastic optimization is a Hamilton--Jacobi- Bellman PDE. We show how this technique (the reduction to a Hamilton--Jacobi--Bellman PDE) allows us to get many recent results in estimating (often sharply) singular integrals of classical type. For example, the solution of $A_2$ conjecture will be given by the manipulations with convex functions of special type, which are the solutions of the corresponding HJB equation.

As an illustration we also compute the numerical value of the norm in $L^p$ of the real and imaginary parts of the Ahlfors--Beurling transform. The upper estimate is again based on a solution of corresponding HJB, which one composes with the heat flow. The estimate from below (we compute the norm, so upper and lower estimates coincide) is obtained by the method of laminates, which is related to a problem of C. B. Morrey from the calculus of variations. We also give a certain (not sharp) estimate of the Ahlfors--Beurling operator itself and explain the connection with Morrey's problem.