Ellipticity and the problem of iterates in Denjoy–Carleman classes

TL;DR

This paper extends a 1978 result on elliptic differential operators to Denjoy–Carleman classes, highlighting a key difference between these and other classes.

Contribution

A new proof technique for Denjoy–Carleman classes and a distinction between their properties and those of Braun–Meise–Taylor classes.

Findings

Ellipticity is characterized by the theorem of iterates in Denjoy–Carleman classes with strongly non-quasianalytic weight sequences.

A novel method constructs optimal functions in Denjoy–Carleman classes using Fourier integrals.

The analogous result does not hold for Braun–Meise–Taylor classes defined by weight functions.

Abstract

In 1978 Métivier showed that a linear differential operator P with analytic coefficients is elliptic if and only if the theorem of iterates holds for P with respect to any non-analytic Gevrey class. In this paper we extend this theorem to Denjoy–Carleman classes given by strongly non-quasianalytic weight sequences. The proof involves a new way to construct optimal functions in Denjoy–Carleman classes via Fourier integrals, which might be of independent interest. Moreover, we point out that the analogous statement for Braun–Meise–Taylor classes given by weight functions cannot hold. This signifies an important difference in the properties of Denjoy–Carleman classes and Braun–Meise–Taylor classes, respectively.