Time-Dependent Loss of Derivatives for Hyperbolic Operators with Non Regular Coefficients

In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension N ≥ 1. 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