On the global solvability in Gevrey classes on the n-dimensional torus

In the last years many papers are concerned with the study of the global solvability and hypoellipticity of linear partial differential operators on compact manifolds, e.g. torus, in large scales of functional spaces. It is well known that the theory of global properties of differential operators is not well developped in comparison with the one of local properties. In particular, the global properties are open problems except for certain operators. On the other hand, the local and global solvability/hypoellipticity are rather different in general. Motivated by these facts, in this paper we are interested in the problem of global solvability for linear partial differential operators in the setting of Gevrey classes $G^s$ of order $s\geq 1$ on the $n$-dimensional torus $T^n$. More precisely, we give a necessary condition for the global solvability in $G^s(T^n)$ in terms of a priori estimate analogous to the ones due to Hormander. The proof of this abstract result is of a functional-analytic nature and more difficult in comparison with the $C^\infty$-case because of the complicated topology of $G^s(T^n)$. We also apply such a necessary condition to give a complete characterization for the Gevrey solvability for a class of linear partial differential operators with variable coefficients, where diophantine properties of the coefficients play a crucial role.