  We consider the Cauchy problem for strictly hyperbolic $m$-th order partial
differential equations with coefficients low-regular in time and smooth in
space. It is well-known that the problem is $L^2$ well-posed in the case of
Lipschitz continuous coefficients in time, $H^s$ well-posed in the case of
Log-Lipschitz continuous coefficients in time (with an, in general, finite loss
of derivatives) and Gevrey well-posed in the case of H\"older continuous
coefficients in time (with an, in general, infinite loss of derivatives). Here,
we use moduli of continuity to describe the regularity of the coefficients with
respect to time, weight sequences for the characterization of their regularity
with respect to space and weight functions to define the solution spaces. We
establish sufficient conditions for the well-posedness of the Cauchy problem,
that link the modulus of continuity and the weight sequence of the coefficients
to the weight function of the solution space. The well-known results for
Lipschitz, Log-Lipschitz and H\"older coefficients are recovered.
