The Green Operator of a Globally Analytic Hypo-elliptic Operator on the Torus and Applications

(joint work with Paulo D.~Cordaro) We study the properties of the Green operator for an analytic linear PDO such that both it and its formal adjoint are globally sub-elliptic and globally analytic-hypoelliptic (GAH) in the torus. We introduce the class of M\'etivier operators, $ \mathscr{M}_{\varepsilon}(\mathbb{T}^{N})$, study the properties of its perturbations and of its analytic vectors and show that when the Green operator of $ P(x,D)$ belongs to a well defined class of analytic pseudodifferential operators on the torus then $ P(x,D) \in \mathscr{M}_{\varepsilon}(\mathbb{T}^{N})$. We present some examples of linear PDO in such class.\\ We also study (joint work with N. ~Braun Rodrigues, Paulo D.~Cordaro and M.~R.~Jahnke) the perturbation problem and the Gevrey regularity of the Gevrey vectors for a class of globally analytic hypoelliptic H\"ormander's operators defined on the $N$-dimensional torus introduced by P.~D.~ Cordaro and A.~A.Himonas.