Counterexamples to C∞ C^{\infty} well posedness for some hyperbolic operators with triple characteristics

In this paper we prove that for a class of non-effectively hyperbolic operators with smooth triple characteristics the Cauchy problem is well posed in the Gevrey 2 class, beyond the generic Gevrey class $3/2$ (see e.g. \cite{Bro}). Moreover we show that this value is optimal