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 $\mathrm{Diff}\mathcal{B}$, $\mathrm{Diff}{W}^{\infty ,p}$, $\mathrm{Diff}\mathcal{S}$, and $\mathrm{Diff}\mathcal{D}$ are $C^{\infty }$ Lie groups, and $\mathrm{Diff}{\mathcal{B}}^{\{M\}}$, $\mathrm{Diff}{W}^{\{M\},p}$, $\mathrm{Diff}{\mathcal{S}}_{\{L\}}^{\{M\}}$, and $\mathrm{Diff}{\mathcal{D}}^{[M]}$, for non-quasianalytic $M$, are $C^{\{M\}}$ Lie groups, where $\mathrm{Diff}\mathcal{A}=\{\mathrm{Id}+f:f\in \mathcal{A}({\mathbb{R}}^n,{\mathbb{R}}^n),{\inf }_{x\in {\mathbb{R}}^n}\det ({\mathbb{I}}_n+df(x))>0\}$. We also discuss stability under composition.

42 pages, mistake corrected, results slightly extended; small changes before publication added. in Indagationes Mathematicae, 2015

References in corpus (1)

• An exotic zoo of diffeomorphism groups on ${\mathbb{R}}^n$

Cited by in corpus (3)

• Manifolds of mappings for continuum mechanics
• The Trouvé group for spaces of test functions
• Interpolation of derivatives and ultradifferentiable regularity