Ellipticity and the problem of iterates in Denjoy-Carleman classes
Stefan Fürdös, Gerhard Schindl
TL;DR
The paper addresses how ellipticity relates to the theorem of iterates for ultradifferentiable classes, extending Métivier's result from analytic/Gevrey settings to Denjoy–Carleman classes defined by strongly non-quasianalytic weight sequences. The authors develop a novel integral construction of optimal functions and prove that, for a non-elliptic operator $P$ with analytic coefficients, there exists a smooth function $u$ with $u \in \mathcal{E}^{(\mathbf{M})}(\Omega;P)$ but $u \notin \mathcal{E}^{\{\mathbf{M}\}}(\Omega)$, while providing a stronger invariant version via the Borel map. They also show that an analogous statement fails for Braun–Meise–Taylor classes, highlighting a fundamental difference between Denjoy–Carleman and BM frameworks, and they develop the existence and construction of optimal $\{\mathbf{N}\}$-flat functions to control the regularity of such vectors. Collectively, these results extend the theory of iterates to a broad Denjoy–Carleman setting, clarify the role of growth indices like $\gamma(\mathbf{M})$, and introduce a versatile integral-method for generating optimal functions with potential applications beyond the present problem.
Abstract
In 1978 Métivier 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.