Double Φ-inequalities for Banach-space-valued martingales

Let B be a Banach space, Φ1, Φ2 be two generalized convex Φ -functions and Ψ1,Ψ2 the Young complementary functions of Φ1, Φ1 respectively with for some constants c0 > 0 and t0 > 0, where ψ11 and ψ2 are the left-continuous derivative functions of Ψ1 and Ψ2, respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f =(fn)n ≥ 0, ‖f*‖Φ1≤c‖S(p)(f)‖Φ2 (or ‖S(q)(f)‖Φ1≤c‖f*‖Φ2, respectively), where f* and S(p)(f) are the maximal function and the p-variation function of f respectively; (ii) If B is a UMD space, Tvf is the martingale transform of f with respect to v=(vn)n≥0(v*≤1), then ‖(Tvf)*‖Φ1≤c‖f*‖Φ2.