Full proof of Kwapie\'n's theorem on representing bounded mean zero functions on $[0,1]$

TL;DR

This paper provides a complete proof of Kwapień's theorem, showing that every bounded mean zero function on [0,1] can be expressed as a coboundary with a measure-preserving transformation, including functions with discontinuities.

Contribution

The authors fill a gap in the original proof for discontinuous functions and extend the result to approximate bounds on the coboundary function g.

Findings

Complete proof of Kwapień's theorem for all bounded mean zero functions.

Extension to approximate bounds on the coboundary function g.

Method applicable to functions with discontinuities.

Abstract

In [7], Kwapie\'{n} announced that every mean zero function $f\in L_\infty[0,1]$ can be written as a coboundary $f = g\circ T -g$ for some $g\in L_\infty[0,1]$ and some measure preserving transformation $T$ of $[0,1]$. Whereas the original proof in [7] 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 Kwapie\'{n}'s result in full generality. Our method also allows to improve the original result by showing that for any given $\epsilon>0$ the function $g$ can be chosen to satisfy a bound $\|g\|_\infty\leq (1+\epsilon)\|f\|_\infty$.