Weak-L∞ inequality for non-symmetric martingale transforms and Haar system

Let b < B be two real numbers. Suppose that f = (fn)n≥0 and g = (gn)n≥0 are two Hilbert-space-valued martingales satisfying an asymmetric (b,B)-differential subordination. The paper contains the proof of the sharp weak-type inequality ||g||{W(Ω)} ≤ 2 max (−b, B)||f||{L∞}, where W is the weak-L∞ space introduced by Bennett, DeVore and Sharpley. As applications, we obtain related estimates for the Haar system and harmonic functions on Euclidean domains.