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

Abstract

In this paper we extend a well-known result concerning hypoellipticity and local solvability of linear partial differential operators on Schwartz distributions (see [14] and [19]) 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 $'T$. As an application, we identify some classes of non-Gevreyhypoelliptic operators. A fundamental kemel is also constructed for any Gevrey hypoelliptic partial differential operator.