Ellipticity and the problem of iterates in Denjoy-Carleman classes

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.