On some inequalities for martingale transforms in Banach function spaces

Let X be a Banach function space over a probability space. We consider the inequality parallel to(v*f)(infinity)parallel to(X) <= C parallel to f(infinity)parallel to(X), where f = (f(n))(n is an element of Z+) is a uniformly integrable martingale, v = (v(n))(n is an element of Z+) is a predictable process such that sup(n) |vn| <= 1 almost surely, and v*f = ((v*f)(n))(n is an element of Z+) denotes the martingale transform of f by v. The main result gives necessary and sufficient conditions on X for this inequality to hold.