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

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 ${\mathrm{Diff }}\mathcal {B}^{[M]}(\mathbb {R}^n)$, ${\mathrm{Diff }}W^{[M],p}(\mathbb {R}^n)$, ${\mathrm{Diff }}{\mathcal {S}}{}_{[L]}^{[M]}(\mathbb {R}^n)$, and ${\mathrm{Diff }}\mathcal {D}^{[M]}(\mathbb {R}^n)$ 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.