An exotic zoo of diffeomorphism groups on Rn

Let be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence is log-convex and has moderate growth. We prove that the groups , , , and of -diffeomorphisms on which differ from the identity by a mapping in (global Denjoy-Carleman), (Sobolev-Denjoy-Carleman), (Gelfand-Shilov), or (Denjoy-Carleman with compact support) are -regular Lie groups. As an application, we use the -transform to show that the Hunter-Saxton PDE on the real line is well posed in any of the classes , , and . Here, we find some surprising groups with continuous left translations and right translations (called half-Lie groups), which, however, also admit R-transforms.