On certain martingale inequalities for maximal functions and mean oscillations - Kikuchi, Masato; Kinoshita, Yasuhiro (MATHEMATICA SCANDINAVICA, 2011)
Let X be a Banach function space over a nonatomic probability space. For a uniformly integrable martingale f=(f_n) with respect to a filtration F=(F_n), let Mf =_n |f_n| and θ_Ff=_n E[|f_∞- f_n-1| F_n]. We give a necessary and sufficient condition on X for the inequality θ_Ff _X ≤ CMf_X to hold.