Isomorphism of Lipschitz-free spaces over Euclidean spaces of different dimensions

Determine whether, for distinct integers n and m strictly greater than 1, the Lipschitz-free Banach spaces F(R^n) and F(R^m) are isomorphic as Banach spaces.

Background

The paper studies relations between several equivalence notions on complete metric spaces, including M ol-equivalence and F-equivalence (isomorphisms of Lipschitz-free spaces). The authors show that M ol-equivalence preserves covering dimension, while F-equivalence does not necessarily do so.

In this context, they point to a longstanding question about Lipschitz-free spaces over Euclidean spaces of differing dimensions greater than one. Their Corollary 4.4 implies that if such an isomorphism existed, it could not preserve the finiteness of supports introduced in [4, Definition 2.5], highlighting the significance of resolving the isomorphism question.