Resumen de On some mean oscillation inequalities for martingales

Masato Kikuchi

• Let (X, k · kX) be a Banach function space over a nonatomic probability space (Ω, Σ, P). If f = (fn)n∈Z+ is a martingale with respect to a filtration F = (Fn)n∈Z+ , then we define θF f = sup 0≤n≤m<∞ E ˆ |fm − fn−1| ˛ ˛ Fn ˜, where f−1 ≡ 0. In this paper, we give a necessary and sufficient condition for the existence of constants c and C such that for any martingale f = (fn)n∈Z+ , c lim n→∞ kfnkX ≤ kθF f kX ≤ C lim n→∞ kfnkX