Smooth Livsic regularity for piecewise expanding maps

We consider the regularity of measurable solutions $\chi$ to the cohomological equation $$\varphi =\chi \circ T-\chi ,$$ where $(T,X,\mu )$ is a dynamical system and $\varphi \text{\hspace{0.17em}⁣}:X\to \mathbb{R}$ is a $C^k$ valued cocycle in the setting in which $T\text{\hspace{0.17em}⁣}:X\to X$ is a piecewise $C^k$ Gibbs–Markov map, an affine $\beta$-transformation of the unit interval or more generally a piecewise $C^k$ uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions $\chi$ possess $C^k$ versions. In particular we show that if $(T,X,\mu )$ is a $\beta$-transformation then $\chi$ has a $C^k$ version, thus improving a result of Pollicott et al.~\cite{Pollicott-Yuri}.