Ellipticity and the problem of iterates in Denjoy-Carleman classes

TL;DR

This paper extends Métivier's theorem linking ellipticity and the theorem of iterates from analytic to Denjoy-Carleman classes with strongly non-quasianalytic weights, highlighting key differences with Braun-Meise-Taylor classes.

Contribution

It generalizes the theorem of iterates to Denjoy-Carleman classes with new optimal function constructions, and contrasts these with Braun-Meise-Taylor classes.

Findings

Theorem of iterates holds for elliptic operators in certain Denjoy-Carleman classes.

Optimal functions in Denjoy-Carleman classes can be constructed using a new method.

The theorem does not extend to Braun-Meise-Taylor classes, indicating fundamental differences.

Abstract

In 1978 M\'etivier showed that a 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, 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.