Relation between the norms of the maximal function and the square function of a martingale

What is it about?

Let X be a Banach function space over a nonatomic probability space Ω and let Mu 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. The author gives 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.

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