On certain martingale inequalities for maximal functions and mean oscillations

Abstract

Let X be a Banach function space over a nonatomic probability space. For a uniformly integrable martingale f = (f n ) with respect to a filtrationWe give a necessary and sufficient condition on X for the inequality θ F f X ≤ C Mf X to hold.

On some martingale inequalities for mean oscillations in weak spaces

“…In [8] it was shown that the inequality θ F f X ≤ C M f X holds for all F = (F n ) ∈ F and all f = ( f n ) ∈ M u (F ) if and only if X can be equivalently renormed so as to be a rearrangement-invariant Banach function space with β X < 1, where C is a constant independent of f , and where β X denotes the upper Boyd index of the renormed space X . It was also shown in [6] that the inequalities…”