Lipschitz-free Spaces Research Articles

Continuous Operators from Spaces of Lipschitz Functions

We study the existence of continuous (linear) operators from the Banach spaces Lip0(M) of Lipschitz functions on infinite metric spaces M vanishing at a distinguished point and from their predual spaces F(M) onto certain Banach spaces, including C(K)-spaces and the spaces c0 and ℓ1. For pairs of spaces Lip0(M) and C(K) we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. It is also showed that if a metric space M contains a bilipschitz copy of the unit sphere Sc0 of the space c0, then Lip0(M) admits a continuous operator onto ℓ1 and hence onto c0. Using this, we provide several conditions for a space M implying that Lip0(M) is not a Grothendieck space. Finally, we obtain a new characterization of the Schur property for Lipschitz-free spaces: a space F(M) has the Schur property if and only if for every complete discrete metric space N with cardinality d(M) the spaces F(M) and F(N) are weakly sequentially homeomorphic.

Tilings of the hyperbolic space and Lipschitz functions

Abstract We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$ , where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$ . In particular, we obtain that, for $d=2,3,4$ , $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$ .

On the strongly subdifferentiable points in Lipschitz-free spaces

On the strongly subdifferentiable points in Lipschitz-free spaces

Lipschitz Operators Associated with Weakly $p$-Compact and Unconditionally $p$-Compact Sets

The study introduces different categories of Lipschitz operators linked with weakly $p$-compact and unconditionally $p$-compact sets. It explores some properties of these operator classes derived from linear operators associated with these sets and examines their interconnections. Additionally, it denotes that these classes are extensions of the related linear operators. Moreover, the study evaluates the concept of majorization by scrutinizing both newly obtained and pre-existing results and draws some conclusions based on these findings. The primary method used to obtain the results in the study is the linearization of Lipschitz operators through the Lipschitz-free space constructed over a pointed metric space.

Lipschitz-free spaces over properly metrizable spaces and approximation properties

Let T T be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrizable space. Let M T \mathcal {M}^T be the non-empty set of all proper metrics d d on T T compatible with its topology, and equip M T \mathcal {M}^T with the topology of uniform convergence, where the metrics are regarded as functions on T 2 T^2 . We prove that the set A T , 1 \mathcal {A}^{T,1} of metrics d ∈ M T d\in \mathcal {M}^T for which the Lipschitz-free space F ( T , d ) \mathcal {F}(T,d) has the metric approximation property is a dense set in M T \mathcal {M}^T , and is furthermore residual in M T \mathcal {M}^T when T T is zero-dimensional. We also prove that if T T is uncountable then the set A f T \mathcal {A}^T_f of metrics d ∈ M T d\in \mathcal {M}^T for which F ( T , d ) \mathcal {F}(T,d) fails the approximation property is dense in M T \mathcal {M}^T . Combining the last statement with a result of Dalet, we conclude that for any ‘properly metrizable’ space T T , A f T \mathcal {A}^T_f is either empty or dense in M T \mathcal {M}^T .

Local complementation in Banach spaces and its preservation under free constructions

In this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation of Banach spaces. As an application, we show that local complementation is naturally preserved under certain free constructions in Functional Analysis, including Lipschitz-free spaces and free Banach lattices.

Convex integrals of molecules in Lipschitz-free spaces

We introduce convex integrals of molecules in Lipschitz-free spaces F(M) as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of F(M) that are induced by Radon measures on M, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of F(M). In particular, we show that all elements of F(M) are convex series of molecules when M is uniformly discrete and identify all extreme points of the unit ball of F(M) in that case.

The numerical index of 2-dimensional Lipschitz-free spaces

We provide the explicit formula for the numerical index of any 2-dimensional Lipschitz-free space, also giving the construction of operators attaining this value as its numerical radius. As a consequence, the numerical index of 2-dimensional Lipschitz-free spaces can take any value of the interval [12,1], and this whole range of numerical indices can be attained by taking 2-dimensional subspaces of any Lipschitz-free space of the form F(A), where A⊂Rn with n≥2 is any set with non-empty interior.

Lipschitz-Free Spaces over Cantor Sets and Approximation Properties

Let K=2^mathbb {N} be the Cantor set, let mathcal {M} be the set of all metrics d on K that give its usual (product) topology, and equip mathcal {M} with the topology of uniform convergence, where the metrics are regarded as functions on K^2. We prove that the set of metrics din mathcal {M} for which the Lipschitz-free space mathcal {F}(K,d) has the metric approximation property is a residual F_{sigma delta } set in mathcal {M}, and that the set of metrics din mathcal {M} for which mathcal {F}(K,d) fails the approximation property is a dense meager set in mathcal {M}. This answers a question posed by G. Godefroy.

Wasserstein distance and metric trees

We study the Wasserstein (or earthmover) metric on the space P(X) of probability measures on a metric space X . We show that, if a finite metric space X embeds stochastically with distortion D in a family of finite metric trees, then P(X) embeds bi-Lipschitz into \ell^1 with distortion D . Next, we re-visit the closed formula for the Wasserstein metric on finite metric trees due to Evans–Matsen (2012).We advocate that the right framework for this formula is real trees, and we give two proofs of extensions of this formula: one making the link with Lipschitz-free spaces from Banach space theory, the other one algorithmic (after reduction to finite metric trees).

Weakly almost square Lipschitz-free spaces

We construct a Lipschitz-free space that is locally almost square but not weakly almost square; this is the first example of such a Banach space. We also prove a result, which indicates that geodesic metric spaces are a potential metric characterization for weakly almost square Lipschitz-free spaces. Lastly, we prove that a Lipschitz-free space can not have the symmetric strong diameter 2 property.

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

We study several properties and applications of the ultrapower MU of a metric space M. We prove that the Lipschitz-free space F(MU) is finitely representable in F(M). We also characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those that biLipschitz embed into an ultrapower of the Banach space. Thanks to this link, we obtain that if M is finitely Lipschitz representable in a Banach space X, then F(M) is finitely representable in F(X). We apply these results to the study of cotype in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces of Lipschitz functions under ultraproducts.

Complementability of isometric copies of ℓ1 in transportation cost spaces

This work aims to establish new results pertaining to the structure of transportation cost spaces. Due to the fact that those spaces were studied and applied in various contexts, they have also become known under different names such as Arens-Eells spaces, Lipschitz-free spaces, and Wasserstein spaces. The main outcome of this paper states that if a metric space X is such that the transportation cost space on X contains an isometric copy of ℓ1, then it contains a 1-complemented isometric copy of ℓ1.

Lipschitz-free spaces over manifolds and the metric approximation property

Let ‖⋅‖ be a norm on RN and let M be a closed C1-submanifold of RN. Consider the pointed metric space (M,d), where d is the metric given by d(x,y)=‖x−y‖, x,y∈M. Then the Lipschitz-free space F(M) has the metric approximation property.

Injectivity of Lipschitz Operators

Any Lipschitz map $$f:M \rightarrow N$$ between metric spaces can be “linearised” in such a way that it becomes a bounded linear operator $$\widehat{f}:\mathcal {F}(M) \rightarrow \mathcal {F}(N)$$ between the Lipschitz-free spaces over M and N. The purpose of this note is to explore the connections between the injectivity of f and the injectivity of $$\widehat{f}$$ . While it is obvious that if $$\widehat{f}$$ is injective then so is f, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces M, any injective Lipschitz map $$f:M \rightarrow N$$ (for any N) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on f which ensure that $$\widehat{f}$$ is injective.

Characterizations of Daugavet points and delta-points in Lipschitz-free spaces

A norm $1$ element $x$ of a Banach space is a Daugavet point (respectively, a $\Delta $-point) if every slice of the unit ball (respectively, every slice of the unit ball containing $x$) contains an element which is at distance almost 2 from $x$. We chara

On large ℓ₁-sums of Lipschitz-free spaces and applications

We prove that the Lipschitz-free space over a Banach space X X of density κ \kappa , denoted by F ( X ) \mathcal {F}(X) , is linearly isomorphic to its ℓ 1 \ell _1 -sum ( ⨁ κ F ( X ) ) ℓ 1 \left (\bigoplus _{\kappa }\mathcal {F}(X)\right )_{\ell _1} . This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 0 over a L p \mathcal {L}_p -space. More precisely, we establish that, for every 1 ≤ p ≤ ∞ 1\leq p\leq \infty , if X X is a L p \mathcal {L}_p -space of density κ \kappa , then L i p 0 ( X ) \mathrm {Lip}_0(X) is either isomorphic to L i p 0 ( ℓ p ( κ ) ) \mathrm {Lip}_0(\ell _p(\kappa )) if p > ∞ p>\infty , or L i p 0 ( c 0 ( κ ) ) \mathrm {Lip}_0(c_0(\kappa )) if p = ∞ p=\infty .

The Lipschitz-Free Space Over a Length Space is Locally Almost Square but Never Almost Square

We prove that the Lipschitz-free space over a metric space M is locally almost square whenever M is a length space. Consequently, the Lipschitz-free space is locally almost square if and only if it has the Daugavet property. We also show that a Lipschitz-free space is never almost square.

Compact and weakly compact Lipschitz operators

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$ . This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.

Stochastic approximation of lamplighter metrics

We observe that embeddings into random metrics can be fruitfully used to study the L 1 $L_1$ -embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L 1 $L_1$ -distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an n $n$ -point metric space embeds bi-Lipschitzly into L 1 $L_1$ with distortion O ( log n ) $O(\log n)$ . In particular, for every finite group G $G$ the lamplighter group H = Z 2 ≀ G $H=\mathbb {Z}_2\wr G$ bi-Lipschitzly embeds into L 1 $L_1$ with distortion O ( log log | H | ) $O(\log \log |H|)$ . In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L 1 $L_1$ of the lamplighter group over the d $d$ -dimensional infinite lattice Z d $\mathbb {Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d $d$ -dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.