Hypoellipticity and local solvability of pseudolocal continuous linear operators in Gevrey classes

In this paper we extend a well-known result concerning hypoellipticity and local solvability of linear partial differential operators on Schwartz distributions to the framework of pseudolocal continuous linear maps T acting on Gevrey classes. Namely we prove that the Gevrey hypoellipticity of T implies the Gevrey local solvability of the transposed operator. As an application, we identify some classes of non-Gevrey-hypoelliptic operators. A fundamental kernel is also constructed for any Gevrey hypoelliptic partial differential operator