Let (Ω,Σ,P) be a complete probability space, F = {F n } ∞ n=0 be an increasing sequence of σ- algebras such that ∪ ∞ n=0 F n generates Σ. If f = {f n } ∞ n=0 is a martingale with respect to F and E n is the conditional expectation with respect to F n , then one can introduce a maximal function M(f) = sup n>0 |f n | and a square function S(f) =?∞P i=0|f i − f i−1 | 2 ¶ 1/2 , f −1 = 0. In the case of uniformly integrable martingales there exists g ∈ L 1 (Ω) such that E n g = f n and we consider a sharp maximal function f ♯ = sup n>0 E n |g − f n−1 |.
