An Exotic Zoo of Diffeomorphism Groups on R

Let C[M ] be a (local) Denjoy–Carleman class of Beurling or Roumieu type, where the weight sequence M = (Mk) is log-convex and has moderate growth. We prove that the groups DiffB[M ](Rn), DiffW [M ],p(Rn), DiffS [M ] [L] (Rn), and DiffD[M ](Rn) of C[M ]-diffeomorphisms on Rn which differ from the identity by a mapping in B[M ] (global Denjoy– Carleman), W [M ],p (Sobolev–Denjoy–Carleman), S [M ] [L] (Gelfand–Shilov), or 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, S [M ] [L] , and 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.