An exotic zoo of diffeomorphism groups on $\mathbb R^n$

TL;DR

This paper establishes that various groups of diffeomorphisms on al R^n, characterized by Denjoy-Carleman classes, are regular Lie groups, and applies this to prove well-posedness of the Hunter-Saxton PDE in these classes.

Contribution

It proves that diffeomorphism groups in Denjoy-Carleman classes are $C^{[M]}$-regular Lie groups and introduces new examples of half-Lie groups with continuous translations.

Findings

Diffeomorphism groups are $C^{[M]}$-regular Lie groups.

Hunter-Saxton PDE is well-posed in specific Denjoy-Carleman classes.

Existence of half-Lie groups with continuous translations and $R$-transforms.

Abstract

Let $C^{[M]}$ be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence $M=(M_k)$ is log-convex and has moderate growth. We prove that the groups ${\operatorname{Diff}}\mathcal{B}^{[M]}(\mathbb{R}^n)$, ${\operatorname{Diff}}W^{[M],p}(\mathbb{R}^n)$, ${\operatorname{Diff}}{\mathcal{S}}{}_{[L]}^{[M]}(\mathbb{R}^n)$, and ${\operatorname{Diff}}\mathcal{D}^{[M]}(\mathbb{R}^n)$ of $C^{[M]}$-diffeomorphisms on $\mathbb{R}^n$ which differ from the identity by a mapping in $\mathcal{B}^{[M]}$ (global Denjoy--Carleman), $W^{[M],p}$ (Sobolev-Denjoy-Carleman), ${\mathcal{S}}{}_{[L]}^{[M]}$ (Gelfand--Shilov), or $\mathcal{D}^{[M]}$ (Denjoy-Carleman with compact support) are $C^{[M]}$-regular Lie groups. As an application we use the $R$-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes $W^{[M],1}$, ${\mathcal{S}}{}_{[L]}^{[M]}$, and $\mathcal{D}^{[M]}$. Here we find some surprising groups with continuous left translations and $C^{[M]}$ right translations (called half-Lie groups), which, however, also admit $R$-transforms.