An exotic zoo of diffeomorphism groups on $\mathbb R^n$
| Autor: | Kriegl, Andreas, Michor, Peter W., Rainer, Armin |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | Ann. Glob. Anal. Geom. 47, 2 (2015), 179-222 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10455-014-9442-0 |
| Popis: | 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. Comment: 45 pages; some small corrections done |
| Databáze: | arXiv |
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