On the Cauchy Problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness

In this paper we prove that for noneffectively hyperbolic operators with smooth double characteristics exhibiting a Jordan block of size 4 on the double manifold the Cauchy problem is well posed in the Gevrey 5 class, beyond the generic Gevrey class 2 ( see e.g. cite{Bro}). Moreover we show that this value is optimal due to some geometric constraints on the Hamiltonian flow of the principal symbol. These results, together with results already proven, give a complete picture of the well posedness of the Cauchy problem around hyperbolic double characteristics.