The authors study the weakly hyperbolic Cauchy problem \[ \partial^2_t u-\sum^n_{k,l=1} a_{kl}(t) \partial^2_{x_k x_l}u+ b(t) u=0,\qquad u(0,x)= u_0(x),\quad \partial_t u(0, x)= u_1(x). \] Here weakly hyperbolic means that \(\sum^n_{k,l=1} a_{k_l}(t) \xi_k\xi_l\geq 0\). It is a delicate problem to prove well-posedness results. If the coefficients are analytic on an interval \([0,T]\), then one has \(C^\infty\) well-posedness. If the coefficients belong to \(C^{k,\alpha}[0, T]\), then one has well-posedness in Gevrey classes of order \(s\leq 1+{k+\alpha\over 2}\). Thus there exists a relation between higher regularity of the time-dependent coefficients on the one hand, and lower regularity of the data on the other hand. The goal of the authors is to fill the gap between analytic regularity and let us say \(C^\infty\)-regularity of the time-dependent coefficients. They define function spaces \(\Gamma(M)([0, T])\) by using the condition \[ |a^{(n)}(t)|\leq C A^nM(n),\quad n\geq 0,\quad t\in [0,T], \] with constants \(C>0\) and \(A\geq 1\). As a special case these classes \(\Gamma(M)([0, T])\) contain the Gevrey spaces. By the aid of \(\Gamma(M)([0, T])\) the authors construct almost optimal classes of well-posedness for the data. These classes are described by the behaviour of the Fourier transform. It turns out that these data classes \(\widehat\Gamma(\Phi)(M(n))\) are contained in \(C^\infty(\mathbb{R}^n)\). Counterexamples show that the results are in some sense sharp. As usually the proof bases on energy estimates but the main work of the authors is to prove the interesting fact that the supposed regularity of the coefficients gives a suitable estimate of the in general bad term \[ \int^T_0 {|a'(t,\xi)|\over a(t,\xi)+ \sigma} dt. \] In a lot of papers a so-called log-condition is assumed to handle this term.