关键词: martingale inequality, square function, maximal function, Orlicz space

Abstract:

Let 1,2 be nonnegative nondecreasing functions, and 1 be concave. The authors prove the equivalence of the following two conditions:
(i) E1(Mf)  cE2(Z0+A1) for every nonnegative submartingale f = (fn)n0 with it’s Doob’s Decomposition: f = Z + A, where Z is a martingale in L1 and A is a nonnegative incrasing and predictable process. (ii) There exists positive constants c, t0 such that R 1t 1(s) s2 ds  c2(t) t , 8t > t0.
When 1 = 2 the condition (ii) above is equivalent to the classical condition p < 1. As a consequence, for a concave function , p < 1 if and only if E1(Mf)  cE2(Z0+A1) for every nonnegative submartingale f.

Key words: martingale inequality, square function, maximal function, Orlicz space