The exponential law for spaces of test functions and diffeomorphism groups

We prove the exponential law $mathcal A(E times F, G) cong mathcal A(E,mathcal A(F,G))$ (bornological isomorphism) for the following classes $mathcal A$ of test functions: $mathcal B$ (globally bounded derivatives), $W^{infty,p}$ (globally $p$-integrable derivatives), $mathcal S$ (Schwartz space), $mathcal D$ (compact sport, $mathcal B^{[M]}$ (globally Denjoy_Carleman), $W^{[M],p}$ (Sobolev_Denjoy_Carleman), $mathcal S_{[L]}^{[M]}$ (Gelfand_Shilov), and $mathcal D^{[M]}$. Here $E, F, G$ are convenient vector spaces (finite dimensional in the cases of $W^{infty,p}$, $mathcal D$, $W^{[M],p}$, and $mathcal D^{[M]})$, and $M=(M_k)$ is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms $operatorname{Diff} mathcal B$, $operatorname{Diff} W^{infty,p}$, $operatorname{Diff} mathcal S$, and $operatorname{Diff}mathcal D$ are $C^infty$ Lie groups, and $operatorname{Diff} mathcal B^{{M}}$, $operatorname{Diff}W^{{M},p}$, $operatorname{Diff} mathcal S_{{L}}^{{M}}$, and $operatorname{Diff}mathcal D^{[M]}$, for non-quasianalytic $M$, are $C^{{M}}$ Lie groups, where $operatorname{Diff}mathcal A = {operatorname{Id} +f : f in mathcal A(mathbb R^n,mathbb R^n), inf_{x in mathbb R^n} det(mathbb I_n+ df(x))>0}$. We also discuss stability under composition.

An exotic zoo of diffeomorphism groups on $mathbb R^n$

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.