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

Abstract

Let f=(fn)n≥0 and g=(gn)n≥0 be two real-Hilbert-space-valued martingales such that (gn)n≥0 is weakly dominated by (fn)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.