WEAK-TYPE INEQUALITY FOR THE MARTINGALE SQUARE FUNCTION AND A RELATED CHARACTERIZATION OF HILBERT SPACES

Let f be a martingale taking values in a Banach space B and let S (f) be its square function. We show that if B is a Hilbert space, then P(S(f) >= 1) <= root e parallel to f parallel to(1) and the constant root e is the best possible. This extends the result of Cox, who established this bound in the real case. Next, we show that this inequality characterizes Hilbert spaces in the following sense: if B is not a Hilbert space, then there is a martingale f for which the above weak-type estimate does not hold.