Survey Article: Bellman function method and sharp inequalities for martingales

Abstract

The Bellman function method is an efficient device which enables relating certain types of estimates arising in probability and harmonic analysis to the existence of the associated special function satisfying appropriate majorization and concavity. This technique has gained considerable interest in recent years and led to many interesting results concerning the boundedness of wide classes of singular integrals, Fourier multipliers, maximal functions and other related objects. The objective of this survey is to describe the Bellman function approach to certain classical results for martingale transforms. We present the detailed study of the weak-type and moment estimates, and develop some arguments which allow us to simplify and extend the statements, originally proven by Burkholder and others.