Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

Abstract

In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties.

Global Gevrey hypoellipticity on the torus for a class of systems of complex vector fields

“…In this work we use the well known characterization of the elements of G s (T n+1 ) by its Fourier coefficients, namely a smooth periodic function f is in G s (T n+1 ) if there exist positive constants C, h and ǫ such that [3,14,15]). We say that α ∈ R m \ Q m is an exponential Liouville vector of order s 1, if there exists ǫ > 0 such that the inequality |qα − p| < e −ǫ|q| 1/s , has infinitely many solutions (p, q) ∈ Z m × Z 1 .…”