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

摘要

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. Weprove 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 amapping 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 toshow that the Hunter-Saxton PDE on the real line is well-posed in any of theclasses $W^{[M],1}$, ${\mathcal{S}}{}_{[L]}^{[M]}$, 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, alsoadmit $R$-transforms.