Exponential laws for weighted function spaces and regularity of weighted mapping groups

Abstract

Let $E$ be a locally convex space, $U\subseteq {\mathbb{R}}^n$ as well as $V\subseteq {\mathbb{R}}^m$ be open and $k,l\in {\mathbb{N}}_0\cup \left\{\infty \right\}$. Locally convex spaces $C^{k,l}(U\times V,E)$ of functions with different degrees of differentiability in the $U$- and $V$-variable were recently studied by H.Alzaareer, who established an exponential law of the form $C^{k,l}(U\times V,E)\cong C^k(U,C^l(V,E))$. We establish an analogous exponential law $C_{{\mathcal{W}}_1\otimes {\mathcal{W}}_2}^{k,l}(U\times V,E)\cong C_{{\mathcal{W}}_1}^k(U,C_{{\mathcal{W}}_2}^l(V,E))$ for suitable spaces of weighted $C^{k,l}$-maps, as well as an analogue for spaces of weighted continuous functions on locally compact spaces. The results entail that certain Lie groups $C_{\mathcal{W}}^l(U,H)$ of weighted mappings introduced by B.Walter are $C^k$-regular, for each $C^k$-regular Lie group $H$ modeled on a locally convex space and a suitable set of weights $\mathcal{W}$.

53 pages. This article is based on the author's master's thesis "Exponential laws for weighted function spaces and regularity of weighted mapping Groups" advised by Helge Glöckner, University of Paderborn, 2015