Factorization in Denjoy-Carleman classes associated to representations of [formula omitted

Abstract

For two types of moderate growth representations of (Rd,+) on sequentially complete locally convex Hausdorff spaces (including F-representations [14]), 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 [14, Conjecture 6.4] for analytic vectors of representations of G=(Rd,+). As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.