Two-Arens-Eells space and bi-Lipschitz tensor product of two metric spaces

Abstract

We introduce the two-Arens-Eells space and the bi-Lipschitz tensor product for two pointed metric spaces, extending the classical Lipschitz tensor product. We show that the space of two-Lipschitz operators admits a canonical linearization through these constructions. We obtain isometric identifications between bi-Lipschitz tensor products and tensor products of Lipschitz-free Banach spaces. As an application, we characterize integral Lip-linear operators in terms of injective bi-Lipschitz tensor norms.