|
Session 17. Functional Analysis: relations to Complex Analysis and PDE
|
An exotic zoo of diffeomorphism groups on \(\mathbb R^n\) |
Armin Rainer, University of Vienna, Austria
|
The talk is based on joint work with Andreas Kriegl and Peter Michor
|
|
|
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.
Let \(\mathcal A\) stand for any of the following classes of mappings:
- \(\mathcal B^{[M]}\) (global Denjoy--Carleman),
- \(W^{[M],p}\) for \(p\ge 1\) (Sobolev--Denjoy--Carleman),
- \(\mathcal S_{[L]}^{[M]}\) (Gelfand--Shilov),
- \(\mathcal D^{[M]}\) (Denjoy--Carleman with compact support).
We prove that the groups \(\operatorname{Diff}\mathcal A(\mathbb R^n)\)
of \(C^{[M]}\)-diffeomorphisms on \(\mathbb R^n\) which differ from the identity by an \(\mathcal A\)-mapping
are \(C^{[M]}\)-regular Lie groups.
As an application we obtain well-posedness of the Hunter--Saxton PDE on the real line in some (extensions)
of the above spaces. Hereby we also find some surprising groups with
continuous left translations and \(C^{[M]}\) right translations.
|
|
| Print version |
|
