On the Maximal Inequalities for Martingales Involving Two Functions

Let Φ(t) and Ψ(t) be nonnegative convex functions, and let φ and ψ be the right continuous derivatives of Φ and Ψ, respectively. In this paper, we prove the equivalence of the following three conditions: (i) ‖f∗‖Φ ≤ c‖f‖Ψ, (ii) LΨ ⊆ HΦ and (iii) ∫ t s0 φ(s) s ds ≤ cψ(ct), ∀t > s0, where LΨ and HΦ are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob’s inequality holds. We also discuss the converse inequalities.