SHARP INEQUALITIES FOR THE HAAR SYSTEM AND MARTINGALE
Abstract: A classical result of Paley and Marcinkiewicz asserts that the Haar system on
forms an unconditional basis in provided . The purpose of the paper is to
study related weak-type inequalities, which can be regarded as a version of this property for
. Probabilistic counterparts, leading to some sharp estimates for martingale transforms,
2000 AMS Mathematics Subject Classification: Primary: 60G42; Secondary:
Keywords and phrases: Haar system, martingale, weak-type inequality, Bellman
function, best constants.