Infinitesimally Lipschitz functions on metric spaces

TL;DR

This paper investigates the space of bounded functions with uniformly bounded infinitesimal Lipschitz constants on metric spaces, comparing it with Lipschitz and Sobolev spaces, and establishing a Banach-Stone theorem in this setting.

Contribution

It introduces and analyzes the space D^{ abla}(X), compares it with Lipschitz and Newtonian spaces, and proves a Banach-Stone theorem for these functions.

Findings

D^{ abla}(X) is compared with abla^{ ext{Lip}}(X) and abla^{ ext{Lip}}(X)

Under doubling measure and Poincaré inequality, D^{ abla}(X) equals N^{1, abla}(X)

A Banach-Stone theorem is established for D^{ abla}(X)

Abstract

For a metric space $X$, we study the space $D^{\infty}(X)$ of bounded functions on $X$ whose infinitesimal Lipschitz constant is uniformly bounded. $D^{\infty}(X)$ is compared with the space $\LIP^{\infty}(X)$ of bounded Lipschitz functions on $X$, in terms of different properties regarding the geometry of $X$. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare $D^{\infty}(X)$ with the Newtonian-Sobolev space $N^{1, \infty}(X)$. In particular, if $X$ supports a doubling measure and satisfies a local Poincar{\'e} inequality, we obtain that $D^{\infty}(X)=N^{1, \infty}(X)$.