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

Abstract

In [7], Kwapień 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 Kwapień'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 $\Vert g{\Vert }_{\infty }\le (1+\epsilon )\Vert f{\Vert }_{\infty }$.

18 pages