  We consider the Cauchy problem for weakly hyperbolic $m$-th order partial
differential equations with coefficients low-regular in time and smooth in
space. It is well-known that in general one has to impose Levi conditions to
get $C^\infty$ or Gevrey well-posedness even if the coefficients are smooth. 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.
Furthermore, we propose a generalized Levi condition that models the influence
of multiple characteristics more freely. We establish sufficient conditions for
the well-posedness of the Cauchy problem, that link the Levi condition as well
as the modulus of continuity and the weight sequence of the coefficients to the
weight function of the solution space. Additionally, we obtain that the
influences of the Levi condition and the low regularity of coefficients on the
weight function of the solution space are independent of each other.
