In this paper we study the well-posedness of weakly hyperbolic systems with
time dependent coefficients. We assume that the eigenvalues are low regular, in
the sense that they are H\"older with respect to t. In the past these kind of
systems have been investigated by Yuzawa \cite{Yu:05} and Kajitani \cite{KY:06}
by employing semigroup techniques (Tanabe-Sobolevski method). Here, under a
certain uniform property of the eigenvalues, we improve the Gevrey
well-posedness result of \cite{Yu:05} and we obtain well-posedness in spaces of
ultradistributions as well. Our main idea is a reduction of the system to block
Sylvester form and then the formulation of suitable energy estimates inspired
by the treatment of scalar equations in \cite{GR:11