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.

45 pages; some small corrections done

References in corpus (1)

• $R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves

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