Connection between global hypoellipticity and global solvability in Gevrey spaces

Let $P$ be a linear partial differential operator with coefficients in the Gevrey class $G^s(T^n)$ where $T^n$ is the $n$-dimensional torus and $s\geq 1$. We announce that if $P$ is $s$-globally hypoelliptic in $T^n$ then its transposed operator $^t P$ is $s$-globally solvable in $T^n$, thus extending to the Gevrey classes the well-known analogous result in the corresponding $C^\infty$ class. We also give other classes of functions fro which such a result holds yet. The proof of these results and several applications will be published elsewhere