  The present paper concerns the well-posedness of the Cauchy problem for
microlocally symmetrizable hyperbolic systems whose coefficients and
symmetrizer are log-Lipschitz continuous, uniformly in time and space
variables. For the global in space problem we establish energy estimates with
finite loss of derivatives, which is linearly increasing in time. This implies
well-posedness in $H^\infty$, if the coefficients enjoy enough smoothness in
$x$. From this result, by standard arguments (i.e. extension and
convexification) we deduce also local existence and uniqueness. A huge part of
the analysis is devoted to give an appropriate sense to the Cauchy problem,
which is not evident a priori in our setting, due to the very low regularity of
coefficients and solutions.
