Full proof of Kwapien's theorem on representing bounded mean zero functions on [0; 1]

Abstract

In 1984, Kwapien announced that every mean zero function f 2 L1[0; 1] can be written as a coboundary f = g o T -g for some g 2 L1[0; 1] and some measure preserving transformation T of [0; 1]. Whereas Kwapien's original proof holds for continuous functions, there is a serious gap in the proof for functions with discontinuities. In this article we fill in this gap and establish Kwapien's result in full generality. Our method also allows us to improve the original result by showing that for any given ϵ > 0 the function g can be chosen to satisfy ∥g∥1 ≤ (1 + ϵ)∥f∥1.