Inequalities for martingale transforms and related characterizations of Hilbert spaces

Suppose that ff is a martingale taking values in a Banach space BB and gg is its transform by a deterministic sequence of numbers in {−1,1}{−1,1}, such that supn‖gn‖≥1supn‖gn‖≥1 almost surely. We show that a certain family of ΦΦ-estimates for ff holds true if and only BB is a Hilbert space.