The exponential law for spaces of test functions and diffeomorphism groups

Abstract

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.