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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 36, Fasc. 1,
pages 147 - 164
 

SHARP INEQUALITIES FOR THE HAAR SYSTEM AND MARTINGALE TRANSFORMS

Adam Osękowski

Abstract: A classical result of Paley and Marcinkiewicz asserts that the Haar system on [0,1] forms an unconditional basis in Lp  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.

2000 AMS Mathematics Subject Classification: Primary: 60G42; Secondary: 60G46.

Keywords and phrases: Haar system, martingale, weak-type inequality, Bellman function, best constants.

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