Speaker
Description
A celebrated result of Burkholder from the 80's identifies the best constant in the $L^p$ estimate for martingale transforms ($1 < p < \infty$). This result is a starting point for numerous extensions and applications in many areas of mathematics. Burkholder's proof exploits the so-called Bellman function method: it rests on the construction of a certain special function, enjoying appropriate size and concavity requirements. This special function is of interest on its own right and appears, quite unexpectedly, in the context of quasiconformal mappings and geometric function theory. There is a dual approach to the $L^p$ bound, invented by Nazarov, Treil and Volberg in the 90's. It gives a slightly worse constant, but the alternative Bellman function plays an independent, significant role in harmonic analysis, as evidenced in many papers in the last 20 years.
The purpose of the talk is to show how to improve the latter approach so that it produces the best constant and to discuss a number of applications.
