Burkholder’s inequalities associated with Orlicz functions in rearrangement invariant spaces

In this paper, we prove martingale inequalities associated with Orlicz functions in the framework of rearrangement invariant spaces. More precisely, let Φ be an Orlicz function and let X be a rearrangement invariant spaces. We establish the new moment Burkholder inequalities when the Simonenko indices of Φ and the Boyd indices of X satisfy some natural conditions. Our approach mainly relies on a new distribution estimate of the Davis type decomposition.