Strong factorization of ultradifferentiable vectors associated with compact Lie group representations

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.

💡 Research Summary

The paper establishes a strong factorization theorem of Dixmier‑Malliavin type for ultradifferentiable vectors associated with representations of compact Lie groups on sequentially complete locally convex Hausdorff spaces. The authors work in the Braun‑Meise‑Taylor framework, fixing a weight function σ that satisfies the standard growth, convexity and non‑quasianalyticity conditions (α)–(δ). Using σ they define Beurling‑type spaces E(σ)(M;E) and Roumieu‑type spaces E{σ}(M;E) of E‑valued ultradifferentiable functions on a real‑analytic manifold M, together with the corresponding spaces of ultradifferentiable vectors for a representation π of a compact Lie group G.

A key technical tool is the “theorem of iterates”: for an elliptic analytic differential operator P on M (for instance the Laplace‑Beltrami operator), the seminorms defined by repeated application of P are equivalent to the seminorms defined by ordinary partial derivatives. This equivalence allows the authors to replace the usual derivative‑based definition of ultradifferentiability by a more intrinsic, representation‑theoretic one.

The main result (Theorem 1.2) states that for any continuous representation π of a compact Lie group G on a Fréchet (or more generally a sequentially complete locally convex) space E, the space E^ω(π) of analytic vectors enjoys the bounded strong factorization property with respect to the convolution algebra A(G) of real‑analytic functions on G. In concrete terms, every analytic vector v can be written as v = Π(χ₁)·Π(χ₂) where χ₁, χ₂ belong to A(G) and Π denotes the natural action of A(G) on vectors via convolution (Π(χ)v = ∫_G χ(x)π(x)v dx). This resolves the conjecture of Gimperlein, Krötz and Lienau (Conjecture 6.4 in their 2016 paper) for compact groups.

The authors go far beyond analytic vectors. For any weight σ they consider the ultradifferentiable vector space E