Sharp inequalities for the Haar system and martingale transforms

Abstrakt

A classical result of Paley and Marcinkiewicz asserts that the Haar system on [0; 1] forms an unconditional basis in L^p provided 1 < p < ∞. The purpose of the paper is to study related weak-type inequalities, which can be regarded as a version of this property for p = 1. Probabilistic counterparts, leading to some sharp estimates for martingale transforms, are presented.