A note on complete hyperbolic operators with log- Zygmund coefficients Ferruccio Colombini, Daniele Del Santo, Francesco Fanelli and Guy Métivier Abstract. The present paper is the continuation of the recent work [7], and it is devoted to strictly hyperbolic operators with non-regular coefficients. We fo- cus here on the case of complete operators whose second order coefficients are log-Zygmund continuous in time, and we investigate the H ∞ well-posedness of the associate Cauchy problem. Mathematics Subject Classification (2000). Primary 35L15; Secondary 35B65, 35S50, 35B45. Keywords. Hyperbolic operators, non-Lipschitz coefficient, log-Zygmund reg- ularity, energy estimates, well-posedness. 1. Introduction The present paper is, in a certain sense, the continuation of the recent work [7]: it is devoted to the study of strictly hyperbolic operators with log-Zygmund in time coefficients. In particular, we want to prove the H ∞ well-posedness of the Cauchy problem related to a complete second order operator, Lu = ∂ 2 t u − N  i,j=1 ∂ i (a ij (t,x) ∂ j u)+ N  j=0 b j (t,x) ∂ j u + c(t,x) u, (1.1) (with ∂ 0 = ∂ t ) whose highest order coefficients satisfy such a regularity assumption (see also relation (2.3) below). The core of the proof is to establish suitable energy estimates for L. It is well-known (see [14]; see also e.g. [13, Ch. IX] or [18, Ch. 6] for analogous results) that, if the coefficients a jk are Lipschitz continuous with respect to t and only measurable in x, then the Cauchy problem for L is well-posed in H 1 × L 2 . If the a jk ’s are Lipschitz continuous with respect to t and C ∞ b (i.e. C ∞ and bounded with all their derivatives) with respect to the space variables, one can recover the