On the maximal inequalities for martingales involving two functions

Abstract

Let Φ(t) and Ψ(t) be nonnegative convex functions, and let φ 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) L Ψ ⊆ H Φ and (iii) ∫ t s0 φ(s)/s ds ≤ cψ(ct), ∀t > s 0 , where L Ψ and H Φ 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.