The present paper is the continuation of the recent work [F. Colombini, D. Del Santo, F. Fanelli and G. Métivier: Time-dependent loss of
derivatives for hyperbolic operators with non-regular coefficients, Comm. Partial Differential Equations 38 (2013), 1791-1817], and it
is devoted to strictly hyperbolic operators with non-regular coefficients. We focus here on the case of complete operators whose second-order coefficients are log-Zygmund continuous in time, and we investigate the
C^\infty well-posedness of the associate Cauchy problem.
