On some inequalities for Doob decompositions in Banach function spaces

Let Phi : R -> [0, infinity) be a Young function and let f = (f(n))(n is an element of Z+) be amartingale such that Phi(f(n)) is an element of L(1) for all n is an element of Z(+). Then the process Phi (f) = (Phi(f(n)))(n is an element of Z+) can be uniquely decomposed as Phi(f(n)) = g(n) + h(n), where g = (g(n))(n is an element of Z+) is a martingale and h = (h(n))(n is an element of Z+) is a predictable nondecreasing process such that h(0) = 0 almost surely. The main results characterize those Banach function spaces X such that the inequality parallel to h(infinity)parallel to(X)<= C parallel to Phi(Mf)parallel to(X) is valid, and those X such that the inequality parallel to h(infinity)parallel to(X) <= C parallel to Phi(Sf)parallel to(X) is valid, where M f and S f denote the maximal function and the square function of f, respectively.