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

TL;DR

This paper develops a strong factorization theorem for ultradifferentiable vectors in certain representations of , solving a conjecture for analytic vectors and impacting the structure of convolution algebras.

Contribution

It introduces Denjoy-Carleman classes for these representations and proves a Dixmier-Malliavin type factorization theorem, including the resolution of a prior conjecture.

Findings

Proves a strong factorization theorem for ultradifferentiable vectors

Solves the conjecture for analytic vectors of  representations

Shows convolution algebras of ultradifferentiable functions satisfy strong factorization

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.