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

TL;DR

This paper extends exponential laws to weighted function spaces, showing that certain weighted mapping groups are regular Lie groups, thus advancing the understanding of their structure and regularity properties.

Contribution

It establishes new exponential laws for weighted function spaces and proves the regularity of weighted mapping groups as Lie groups.

Findings

Weighted exponential laws analogous to classical ones are proven.

Weighted mapping groups are shown to be $C^k$-regular Lie groups.

Results apply to spaces of weighted continuous functions on locally compact spaces.

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 $C^{k,l}_{\mathcal{W}_1\otimes\mathcal{W}_2}(U\times V,E)\cong C^k_{\mathcal{W}_1}(U,C^l_{\mathcal{W}_2}(V,E))$ 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 $C^l_\mathcal{W}(U,H)$ 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 $\mathcal{W}$.