  Oseledets regularity functions quantify the deviation of the growth
associated with a dynamical system along its Lyapunov bundles from the
corresponding uniform exponential growth. Precise degree of regularity of these
functions is unknown. We show that for every invariant Lyapunov bundle of a
volume preserving Anosov flow on a closed smooth Riemannian manifold, the
corresponding Oseledets regularity functions are in $L^p(m)$, for some $p > 0$,
where $m$ is the probability measure defined by the volume form. We prove an
analogous result for essentially bounded cocycles over volume preserving Anosov
flows.
