Factorization in Denjoy-Carleman classes associated to representations of $(\mathbb{R}^{d},+)$

Abstract

For two types of moderate growth representations of $(\mathbb{R}^d,+)$ on sequentially complete locally convex Hausdorff spaces (including F-representations [J. Funct. Anal. 262 (2012), 667-681], we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [Conjecture 6.; J. Funct. Anal. 262 (2012), 667-681] for analytic vectors of representations of $G =(\mathbb{R}^d,+)$. As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.