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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 31, Fasc. 2,
pages 227 - 238
 

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

Adam Osękowski

Abstract: 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) ≤ √e∥f∥ 1
and the constant √- 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.

2000 AMS Mathematics Subject Classification: Primary: 60G42; Secondary: 46C15.

Keywords and phrases: Martingale, square function, weak type inequality, Banach space, Hilbert space.

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