The Convenient Setting for Quasianalytic Denjoy--Carleman Differentiable Mappings

TL;DR

This paper establishes that for certain quasianalytic Denjoy--Carleman classes, the category of differentiable mappings is cartesian closed, enabling a robust functional framework with applications to diffeomorphism groups.

Contribution

It proves the cartesian closedness of $C^Q$-mapping categories for specific quasianalytic classes, extending the functional analytic structure in this setting.

Findings

The category of $C^Q$-mappings is cartesian closed.

The group of $C^Q$-diffeomorphisms forms a regular $C^Q$-Lie group.

Applications to manifolds of mappings are demonstrated.

Abstract

For quasianalytic Denjoy--Carleman differentiable function classes $C^Q$ where the weight sequence $Q=(Q_k)$ is log-convex, stable under derivations, of moderate growth and also an $\mathcal L$-intersection (see 1.6), we prove the following: The category of $C^Q$-mappings is cartesian closed in the sense that $C^Q(E,C^Q(F,G))\cong C^Q(E\times F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^Q$-diffeomorphisms is a regular $C^Q$-Lie group but not better.