Equivalence of extremal problems on BMO and estimates of martingale transform of bounded martingales

The purpose of the talk is to study the interplay between
the following two extremal problems, arising in the context of analysis
and probability:

1) Find the supremum of an f-functional over the class of all BMO
functions with prescribed averages.

2) Find the supremum of an f-functional over the class of martingale
transforms of bounded martingales, with prescribed averages.

These two absolutely different problems appear to be equivalent: for any
given functional f, the solution each of them can be easily rewritten into
the solution of the other one.

This is a joint result with D. Stolyarov and P. Zatitskii.