Isomorphisms between spaces of Lipschitz functions

Abstract

We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that ${\mathrm{Lip}}_0({\mathbb{Z}}^d)\simeq {\mathrm{Lip}}_0({\mathbb{R}}^d)$, for all $d\in \mathbb{N}$. More generally, we e.g. show that ${\mathrm{Lip}}_0(\Gamma )\simeq {\mathrm{Lip}}_0(G)$, where $\Gamma$ is from a large class of finitely generated nilpotent groups and $G$ is its Mal'cev closure; or that ${\mathrm{Lip}}_0({\ell }_p)\simeq {\mathrm{Lip}}_0(L_p)$, for all $1\le p<\infty$. We leave a large area for further possible research.

arXiv comment28 pages, no figures. Accepted to Journal of Functional Analysis

References in corpus (8)

• On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry
• A fixed point theorem for L^1 spaces
• On the structure of Lipschitz-free spaces
• Differentiating maps into L^1 and the geometry of BV functions
• The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of ${\mathbb{R}}^N$
• Automorphism groups of simplicial complexes and rigidity for uniformly bounded representations
• Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete
• Free spaces over some proper metric spaces

Cited by in corpus (8)

• Lipschitz free spaces isomorphic to their infinite sums and geometric applications
• Approximation properties in Lipschitz-free spaces over groups
• On the geometry of Banach spaces of the form ${\mathrm{Lip}}_0(C(K))$
• Projecting Lipschitz functions onto spaces of polynomials
• Hyperbolic Metric Spaces and Stochastic Embeddings
• Projections in Lipschitz-free spaces induced by group actions
• Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces
• Lipschitz spaces over non-porous sets