Weak-[formula omitted] inequality for non-symmetric martingale transforms and Haar system

Abstract

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 |dgn−B+b2dfn|≤|B−b2dfn|,n=0,1,2,….The paper contains the proof of the sharp weak-type inequality ‖g‖W(Ω)≤2max(−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.