A characterization of Banach function spaces associated with martingales

Let (Omega, Sigma, P) be a nonatomic probability space and let F= (F-n)(nis an element ofZ+) be a filtration. If f = (f(n))(nis an element ofZ+) is a uniformly integrable F-martingale, let A(F)f = denote the martingale defined byA(F)f(n) = E[\finfinity\] (nis an element ofZ(+)), where finfinity = lim(n)f(n) a.s. Let X be a Banach function space over Omega. We give a necessary and sufficient condition for X to have the property that S(f) is an element of X if and only if S(A(F)f) is an element of X, where S(f) stands for the square function of f = (f(n)). 2000 Mathematical Subject Classification.