A Relation between two Kinds of Norms for Martingales

Abstract

Abstract Let X be a Banach function space over a nonatomic probability space Ω and let M- denote the collection of all uniformly integrable martingales on Ω. For f = (fn)n∈Z+ ∈Mu, let Mf denote the maximal function of f, and let f∞ denote the almost sure limit of f. We give some necessary and sufficient conditions for X to have the property that if f, g ∈Mu and ‖Mg‖x ≤ ‖Mf‖x, then ‖g∞‖X ≤ C‖f∞‖x.