We are concerned with the problem of determining the sharp regularity of the coefficients with respect to the time variable t in order to have a well-posed Cauchy problem in H ∞ or in Gevrey classes for linear or quasilinear hyperbolic operators of higher order. We use and mix two different scales of regularity of global and local type: the modulus of Hölder continuity and/or the behaviour with respect to |t − t1| −q , q ≥ 1, of the first derivative as t tends to a point t1. Both are ways to weaken the Lipschitz regularity.