We provide a simplified proof of the existence, under some assumptions, of a
spectral gap for the Perron-Frobenius operator of piecewise uniformly expanding
maps on Riemannian manifolds when acting on some Sobolev spaces. Its
consequences include, among others, the existence of invariant physical
measures, and an exponential decay of correlations for suitable observables.
These features are then adapted to different function spaces (functions with
bounded variation or bounded oscillation), so as to give a new insight of - and
generalize - earlier results.