Biseparating maps between Lipschitz function spaces

TL;DR

This paper characterizes linear biseparating maps between Lipschitz function spaces on complete metric spaces, showing they imply bi-Lipschitz homeomorphisms and exploring their automatic continuity and scalar-valued cases.

Contribution

It provides a comprehensive description of biseparating maps between Lipschitz function spaces, including conditions for automatic continuity and scalar-valued function space characterizations.

Findings

X and Y are bi-Lipschitz homeomorphic under such maps

Automatic continuity of biseparating maps is established in some cases

Characterization of separating bijections for scalar-valued Lipschitz functions when Y is compact

Abstract

For complete metric spaces $X$ and $Y$, a description of linear biseparating maps between spaces of vector-valued Lipschitz functions defined on $X$ and $Y$ is provided. In particular it is proved that $X$ and $Y$ are bi-Lipschitz homeomorphic, and the automatic continuity of such maps is derived in some cases. Besides, these results are used to characterize the separating bijections between scalar-valued Lipschitz function spaces when $Y$ is compact.