A weak-L∞ inequality for weakly dominated martingales with applications to Haar shift operators

Abstract Let f = ( f n ) n ≥ 0 and g = ( g n ) n ≥ 0 be two real-Hilbert-space-valued martingales such that ( g n ) n ≥ 0 is weakly dominated by ( f n ) n ≥ 0 . The paper contains the proof of the inequality ‖ g ‖ W ( Ω ) ≤ 6 ‖ f ‖ L ∞ , where W is the weak- L ∞ space introduced by Bennett, DeVore and Sharpley. As an application, a related estimate for Haar shift operators is established.