  In this paper we will study the Cauchy problem for strictly hyperbolic
operators with low regularity coefficients in any space dimension $N\geq1$. We
will suppose the coefficients to be log-Zygmund continuous in time and
log-Lipschitz continuous in space. Paradifferential calculus with parameters
will be the main tool to get energy estimates in Sobolev spaces and these
estimates will present a time-dependent loss of derivatives.
