  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
$\phi \colon X\rightarrow \R$ is a $C^k$ 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 et al.~\cite{Pollicott-Yuri}.
