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arXiv · 2411.13538

An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces

Gonzalo Flores
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Abstract

For a domain $\Omega$ in a finite-dimensional space $E$, we consider the space $M=(\Omega,d)$ where $d$ is the intrinsic distance in $\Omega$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of $L^{\infty}(\Omega;E^{*})$ and we use this representation in order to obtain the corresponding isometric representation for the Lipschitz-free space $\mathcal{F}(M)$ as a quotient of the space $L^{1}(\Omega;E)$. We compare our result with those existent in the literature for bounded domains with Lipschitz boundary, and for convex domains, which can be then deduced as a corollaries of our result.

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Gonzalo Flores. 2024-11-20. An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces. https://arxiv.org/abs/2411.13538

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