Smooth Livšic regularity for piecewise expanding maps

We consider the regularity of measurable solutions $\chi$ to the cohomological equation

$$\phi = \chi \circ T -\chi,$$

where $(T,X,\mu)$ is a dynamical system and $\phi \colon X\rightarrow \mathbb{R}$ is a $C^k$ smooth real-valued cocycle in the setting in which $T \colon X\rightarrow X$ 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 and Yuri.