Lipschitz free spaces on finite metric spaces

TL;DR

This paper investigates the structure of Lipschitz free spaces on finite metric spaces and certain graph families, revealing their relation to  spaces and highlighting the complexity of their isomorphic properties.

Contribution

It establishes the presence of large -like subspaces in Lipschitz free spaces on finite metric spaces and shows that many recursive graph families are not uniformly -isomorphic.

Findings

Lipschitz free spaces on finite metric spaces contain large well-complemented ^n subspaces.

Lipschitz free spaces on certain recursive graphs are not uniformly isomorphic to ^n.

The approach involves averaging over cycle-preserving bijections and projection constant estimates.

Abstract

Main results of the paper: (1) For any finite metric space $M$ the Lipschitz free space on $M$ contains a large well-complemented subspace which is close to $\ell_1^n$. (2) Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of graphs which are not necessarily graph automorphisms; (b) In the case of such recursive families of graphs as Laakso graphs we use the well-known approach of Gr\"unbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.