On the Bellman function method for operators on martingales

It is shown how to apply the Bellman function method to general operators on martingales, i.e., to operators that are not necessarily martingale transforms. As examples of such operators, we consider the Haar transforms and an operator whose $L^p$-boundedness implies the Rubio de Francia inequality for the Walsh system. For the corresponding Bellman function, the Bellman induction is carried out and a Bellman candidate is constructed.