Annals of Global Analysis and Geometry

, Volume 47, Issue 2, pp 179–222 | Cite as

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

Article

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.

Keywords

Diffeomorphism groups Convenient setting Ultradifferentiable test functions Sobolev Denjoy–Carleman classes Gelfand–Shilov classes Hunter–Saxton equation 

Mathematics Subject Classification

26E10 46A17 46E50 46F05 58B10 58B25 58C25 58D05 58D15 35Q31 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Andreas Kriegl
    • 1
  • Peter W. Michor
    • 1
  • Armin Rainer
    • 1
  1. 1.Fakultät für MathematikUniversität WienWienAustria

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