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

1512.07211Dec 22, 201553 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

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

Natalie Nikitin

Functional Analysis

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Let $E$ be a locally convex space, $U\subseteq\mathbb{R}^n$ as well as $V\subseteq\mathbb{R}^m$

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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}

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