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Strong factorization of ultradifferentiable vectors associated with compact Lie group representations

Andreas Debrouwere, Michiel Huttener, Jasson Vindas

TL;DR

This work develops a comprehensive factorization theory for ultradifferentiable vectors in representations of compact Lie groups on sequentially complete locally convex spaces. By formulating vector-valued Beurling–Roumieu ultradifferentiable classes via weight functions and leveraging the Laplace–Beltrami operator and the Peter–Weyl Fourier framework, the authors reduce factorization to a discrete Fourier coefficient division problem and prove a bounded strong factorization property: $E^{[oldsymbol{ au}]} = oldsymbol{ abla}^{[oldsymbol{ au}]}(G)E^{[oldsymbol{ au}]}$. This generalizes the classical Dixmier–Malliavin factorization to ultradifferentiable settings and resolves the Gimperlein–Krötz–Lienau conjecture for analytic vectors on compact groups, while also providing non-quasianalytic refinements and connections to analytic and Gevrey contexts. The results yield a unified approach to factorization across analytic, Gevrey, and general ultradifferentiable regimes with potential applications in representation theory and harmonic analysis on Lie groups.

Abstract

We show a strong factorization theorem of Dixmier-Malliavin type for ultradifferentiable vectors associated with compact Lie group representations on sequentially complete locally convex Hausdorff spaces. In particular, this solves a conjecture by Gimperlein et al. [J. Funct. Anal. 262 (2012), 667-681] for analytic vectors in the case of compact Lie groups.

Strong factorization of ultradifferentiable vectors associated with compact Lie group representations

TL;DR

This work develops a comprehensive factorization theory for ultradifferentiable vectors in representations of compact Lie groups on sequentially complete locally convex spaces. By formulating vector-valued Beurling–Roumieu ultradifferentiable classes via weight functions and leveraging the Laplace–Beltrami operator and the Peter–Weyl Fourier framework, the authors reduce factorization to a discrete Fourier coefficient division problem and prove a bounded strong factorization property: . This generalizes the classical Dixmier–Malliavin factorization to ultradifferentiable settings and resolves the Gimperlein–Krötz–Lienau conjecture for analytic vectors on compact groups, while also providing non-quasianalytic refinements and connections to analytic and Gevrey contexts. The results yield a unified approach to factorization across analytic, Gevrey, and general ultradifferentiable regimes with potential applications in representation theory and harmonic analysis on Lie groups.

Abstract

We show a strong factorization theorem of Dixmier-Malliavin type for ultradifferentiable vectors associated with compact Lie group representations on sequentially complete locally convex Hausdorff spaces. In particular, this solves a conjecture by Gimperlein et al. [J. Funct. Anal. 262 (2012), 667-681] for analytic vectors in the case of compact Lie groups.

Paper Structure

This paper contains 11 sections, 27 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Spaces of vector-valued ultradifferentiable functions
  3. Weight functions
  4. Ultradifferentiable classes on open subsets of Rⁿ
  5. Ultradifferentiability on manifolds
  6. The theorem of iterates
  7. Ultradifferentiable vectors and the factorization theorem
  8. Ultradifferentiability via the Laplace-Beltrami operator
  9. Characterizing ultradifferentiable functions via their Fourier coefficients
  10. Proof of the main result
  11. Factorization of non-quasianalytic vectors

Key Result

Theorem 1.2

Let $\pi$ be a representation of a compact Lie group $G$ on a Fréchet space $E$. Then, $E^{\omega}$ has the bounded strong factorization property with respect to the convolution algebra of real analytic functions on $G$.

Theorems & Definitions (57)

  • Conjecture 1.1: G-K-L
  • Theorem 1.2
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 47 more