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 LipMore generally, we e.g. show that Lip 0 (Γ) ≃ Lip 0 (G), where Γ is from a large class of finitely generated nilpotent groups and G is its Mal'cev closure; or that Lip 0 (ℓp) ≃ Lip 0 (Lp), for all 1 ≤ p < ∞.We leave a large area for further possible research.2010 Mathematics Subject Classification. 46B03 (primary), and 22E40 (secondary).

Lipschitz free spaces isomorphic to their infinite sums and geometric applications

“…These include Lee and Naor's result [33] on the existence of K-random partitions of unity with respect to any subspace of a doubling metric space, Kalton's result from [28] that every Lipschitz-free space embeds into the infinite direct hand, isomorphic to its 1 -sum and, on the other hand, isomorphic to the Lipschitz-free space over its unit ball. We also extend the results from the paper [12], which are formulated only in terms of the duals of Lipschitz-free spaces. Not only did we succeed to prove analogues of all these results for Lipschitz free p-spaces with p ∈ (0, 1] but, revisiting the topic, we actually found new results and applications which are also of interest for the classical case p = 1.…”

“…Since rescaling of the metric space gives isometric Lipschitz free spaces, we are done Corollary 5.10. Let M be a pointed doubling self-similar metric space and let N ⊂ M be a net in M. Then, for every p ∈ (0, 1], we have [12,Corollary 1.18]). Thus, by Theorem 5.8, F p (N 0 ) p (F p (N 0 )).…”

“…Let f : M → M be the bijection from the definition of a selfsimilar space and let 0 ∈ M be such that f (0) = 0. It is not difficult to construct a 1-separated and 1-dense set N 0 ⊂ M for which f (N 0 ) ⊂ N 0 (see, e.g., the proof of [12,Corollary 1.18]). For n ∈ N put X n = [δ M (f −n (x)) : x ∈ N 0 ].…”

“…The following is an improvement of[12, Corollary 1.18].Corollary 5.11. Let (M, d, 0) be a pointed doubling self-similar metric space and let N ⊂ M be a net in M. Then Lip 0 (N ) Lip 0 (M).Proof.…”

Projecting Lipschitz functions onto spaces of polynomials

“…The study of Banach spaces of Lipschitz functions, and even more of their canonical preduals, the Lipschitz-free spaces, has been one of the most active fields of research within Banach space theory in the last two decades. We refer, e.g., to [AACD, AlP, GK, Go] and [CCD,CK,We,We1] for recent results and additional references on Lipschitz-free spaces and spaces of Lipschitz functions, respectively. Starting from the result [GK] that a Banach space X has the bounded approximation property if and only if the Lipschitz-free space F (X) over X has it, approximation properties of Lipschitz-free Banach spaces have been widely investigated, [AmP, GO, Go, LP, PS].…”