On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness

For a hyperbolic second-order differential operator , we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and some algebraic and geometric properties of the principal symbol $p$. If the Hamilton map $F_{p}$ of $p$ (the linearization of the Hamilton field $H_{p}$ along double characteristics) has nonzero real eigenvalues at every double characteristic (the so-called effectively hyperbolic case), then it is well known that the Cauchy problem for $P$ is well posed in any Gevrey class $1leq stextless +infty$ for any lower-order term. In this paper we prove that if $p$ is noneffectively hyperbolic and, moreover, such that $operatorname{Ker}F_{p}^{2}cap operatorname{Im}F_{p}^{2}neq{0}$ on a $C^{infty}$ double characteristic manifold $Sigma$ of codimension $3$, assuming that there is no null bicharacteristic landing $Sigma$ tangentially, then the Cauchy problem for $P$ is well posed in the Gevrey class $1leq stextless 4$ for any lower-order term (strong Gevrey well-posedness with threshold $4$), extending in particular via energy estimates a previous result of Hörmander in a model case.