Infinitesimally Lipschitz functions on metric spaces

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{é} inequality, we obtain that $D^{\infty }(X)=N^{1,\infty }(X)$.

28 pages, 2 figures