Carleson measures and vector-valued BMO martingales

We study the relationship between vector-valued BMO martingales and Carleson measures. Let (Omega, F, P) be a probability space and 2 <= q < infinity. Let X be a Banach space. Given a stopping time tau, let <(tau)over cap> denote the tent over tau: (tau) over cap = {(w, k) is an element of Omega x N : tau(w) <= k, tau(w) < infinity}. We prove that there exists a positive constant c such that sup(tau) 1/P(tau <infinity)integral((tau) over cap) parallel to dfk parallel to(q)dp circle times dm <= c(q) parallel to f parallel to(q)(B M O(X)) for any finite martingale with values in X iff X admits an equivalent norm which is q-uniformly convex. The validity of the converse inequality is equivalent to the existence of an equivalent p-uniformly smooth norm. And then we also give a characterization of UMD Banach lattices.