Let f, g be two Hilbert-space-valued martingales such that g is differentially subordinate to f. The paper contains the proof of the estimate ‖g‖p,∞≤2p(p+1)p−1‖f‖p,∞,1p∞. The constant is shown to be of optimal order for p→∞ and for p→1. Related results for transforms of...

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