Lipschitz-Free Spaces over Manifolds and the Metric Approximation Property

TL;DR

This paper proves that Lipschitz-free spaces over closed $C^1$-submanifolds of Euclidean space possess the Metric Approximation Property, advancing understanding of their geometric and functional analytic structure.

Contribution

It establishes that Lipschitz-free spaces over certain manifolds have the Metric Approximation Property, a significant result in the theory of Lipschitz and Banach spaces.

Findings

Lipschitz-free spaces over closed $C^1$-submanifolds have the Metric Approximation Property.

The result applies to manifolds embedded in Euclidean space with the induced metric.

This advances the understanding of the structure of Lipschitz-free spaces over manifolds.

Abstract

Let $\|\cdot\|$ be a norm on $\mathbb{R}^N$ and let $M$ be a closed $C^1$-submanifold of $\mathbb{R}^N$. Consider the pointed metric space $(M,d)$, where $d$ is the metric given by $d(x,y)=\|x-y\|$, $x,y\in M$. Then the Lipschitz-free space $\mathcal{F}(M)$ has the Metric Approximation Property.