WEAK-TYPE INEQUALITY FOR THE MARTINGALE SQUARE
FUNCTION AND A RELATED CHARACTERIZATION OF HILBERT
SPACES
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
![ℙ(S(f) ≥ 1) ≤ √e∥f∥
1](files/31.2/HTML/31.2.4.abs4x.png)
and
the constant
![√-
e](files/31.2/HTML/31.2.4.abs5x.png)
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](files/31.2/HTML/31.2.4.abs6x.png)
is not a Hilbert space, then there is a martingale
![f](files/31.2/HTML/31.2.4.abs7x.png)
for which the
above weak-type estimate does not hold.
2000 AMS Mathematics Subject Classification: Primary: 60G42; Secondary:
46C15.
Keywords and phrases: Martingale, square function, weak type inequality, Banach
space, Hilbert space.