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 [15] and Kajitani [14] by employing semigroup
techniques (Tanabe-Sobolevski method). Here, under a certain uniform property of
the eigenvalues, we improve the Gevrey well-posedness result of [15] 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.