WEAK-TYPE INEQUALITY FOR THE MARTINGALE SQUARE
FUNCTION AND A RELATED CHARACTERIZATION OF HILBERT
Abstract: Let be a martingale taking values in a Banach space and let be its square
function. We show that if is a Hilbert space, then
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
is not a Hilbert space, then there is a martingale
for which the
above weak-type estimate does not hold.
2000 AMS Mathematics Subject Classification: Primary: 60G42; Secondary:
Keywords and phrases: Martingale, square function, weak type inequality, Banach
space, Hilbert space.