Infinitesimally Lipschitz functions on metric spaces

Details Infinitesimally Lipschitz functions on metric spaces Infinitesimally Lipschitz functions on metric spaces

2009-01-21

arxiv.org/abs/0901.3236v1

Mathematics

Metric Geometry

28 pages, 2 figures

Scientific paper

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 $D^{\infty}(X)=N^{1, \infty}(X)$.

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