The approximation property for spaces of Lipschitz functions

On projectional skeletons and the Plichko property in Lipschitz-free Banach spaces

We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all $\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\ell_\infty$ contain…

Some stability properties for the Bishop--Phelps--Bollobás property for Lipschitz maps

We study the stability behavior of the Bishop-Phelps-Bollobás property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollobás property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very closed to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property fr…

Lipschitz-Free Spaces over Manifolds and the Metric Approximation Property

Let $\|\cdot\|$ be a norm on $\mathbb{R}^N$ and let $M$ be a closed $C^1$-submanifold of $\mathbb{R}^N$. Consider the pointed metric space $(M,d)$, where $d$ is the metric given by $d(x,y)=\|x-y\|$, $x,y\in M$. Then the Lipschitz-free space $\mathcal{F}(M)$ has the Metric Approximation Property.

A characterisation of the Daugavet property in spaces of Lipschitz functions

We study the Daugavet property in the space of Lipschitz functions $\operatorname{Lip}_0(M)$ for a complete metric space $M$. Namely we show that $\operatorname{Lip}_0(M)$ has the Daugavet property if and only if $M$ is a length space. This condition also characterises the Daugavet property in the Lipschitz free space $\mathcal{F}(M)$. Moreover, when $M$ is compact, we show that either $\mathcal{F}(M)$ has the Daugavet property or its unit ball has a strongly exposed point. If $M$ is an infinite compact subset of a strictly convex Banach space then the Daugavet property of $\operatorname{Lip}_…

关于在Lipschitz-free空间中实现直径二性质的注记

A note on attaining diameter two properties in Lipschitz-free spaces

本文证明在Lipschitz-free空间中，强直径二性质、直径二性质和局部直径二性质与其对应的实现变体一致。

Diameter two properties for spaces of Lipschitz functions

We solve some open problems regarding diameter two properties within the class of Banach spaces of real-valued Lipschitz functions by using the de Leeuw transform. Namely, we show that: the diameter two property, the strong diameter two property, and the symmetric strong diameter two property are all different for these spaces of Lipschitz functions; the space $\operatorname{Lip}_0(K_n)$ has the symmetric strong diameter two property for every $n\in \mathbb{N}$, including the case of $n=2$; every local norm-one Lipschitz function is a Daugavet point.

The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$

We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular, $\mathcal{F}(M)$ has the MAP whenever $M$ is a finite-dimensional compact convex set. This should be compared with a recent result of Godefroy and Ozawa, who showed that there exists a compact convex subset $M$ of a separable Banach space, for which $\mathcal{F}(M)$ fails the approximation property.

A characterisation of octahedrality in Lipschitz-free spaces

We characterise the octahedrality of Lipschitz-free space norm in terms of a new geometric property of the underlying metric space. We study the metric spaces with and without this property. Quite surprisingly, metric spaces without this property cannot embed isometrically into $\ell_1$ and similar Banach spaces.

The Bishop--Phelps--Bollobás property for Lipschitz maps

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness assumptions are needed.…

佩尔切尼茨的属性(V$^*$)在利普希茨自由空间中

Pełczyński's property (V$^*$) in Lipschitz-free spaces

本文证明了利普希茨自由空间中佩尔切尼茨属性(V$^*$)的局部决定性，并获得若干充分条件。证明当完全度量空间M局部紧致且纯1-不可测，或为希尔伯特空间，或属于满足双霍尔德条件的卡诺-卡拉泰多里空间类时，F(M)具有属性(V$^*$)。

The Daugavet property for spaces of Lipschitz functions

For a compact metric space $K$ the space $\Lip(K)$ has the Daugavet property if and only if the norm of every $f \in \Lip(K)$ is attained locally. If $K$ is a subset of an $L_p$-space, $1<p<\infty$, this is equivalent to the convexity of $K$.

Lipschitz-free spaces and Schur properties

In this paper we study $\ell_1$-like properties for some Lipschitz-free spaces. The main result states that, under some natural conditions, the Lipschitz-free space over a proper metric space linearly embeds into an $\ell_1$-sum of finite dimensional subspaces of itself. We also give a sufficient condition for a Lipschitz-free space to have the Schur property, the $1$-Schur property and the $1$-strong Schur property respectively. We finish by studying those properties on a new family of examples, namely the Lipschitz-free spaces over metric spaces originating from $p$-Banach spaces, for $p$ in…

Weak compactness and fixed point property for affine bi-Lipschitz maps

Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz affine mappings for every $L>1$. It is also proved that if $X$ has Pełczyński's property $(u)$, then either $C$ is weakly compact, contains an $\ell_1$-sequence or a $\mathrm{c}_0$-summing basic sequence. In this case, weak compactness of $C$ is equivalent to the $\mathcal{G}$-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We…

The BSE property for vector-valued Frechet Lipschitz algebras

Let $( X,d )$ be a metric space with at least two elements and $( A , p_l )$ be a commutative semisimple Frechet algebra over the scalar field of complex numbers. The correlation between the BSE-property of the Frechet algebra $( A , p_l )$ and $\Lip_{d}( X , A )$ is assessed. It is found and approved that if $\Lip_{d}( X , A )$ is a BSE-Frechet algebra, then so is $A$. The opposite correlation will hold if $( A , p_l )$ is unital.

The Daugavet property in spaces of vector-valued Lipschitz functions

We prove that if a metric space $M$ has the finite CEP then $\mathcal F(M)\widehat{\otimes}_π X$ has the Daugavet property for every non-zero Banach space $X$. This applies, for instance, if $M$ is a Banach space whose dual is isometrically an $L_1(μ)$ space. If $M$ has the CEP then $L(\mathcal F(M),X)=\Lip(M,X)$ has the Daugavet property for every non-zero Banach space $X$, showing that this is the case when $M$ is an injective Banach space or a convex subset of a Hilbert space.

Selection type results and fixed point property for affine bi-Lipschitz maps

We obtain a refinement of a selection principle for $(\mathcal{K}, λ)$-wide-$(s)$ sequences in Banach spaces due to Rosenthal. This result is then used to show that if $C$ is a bounded, non-weakly compact, closed convex subset of a Banach space $X$, then there exists a Hausdorff vector topology $τ$ on $X$ which is weaker than the weak topology, a closed, convex $τ$-compact subset $K$ of $C$ and an affine bi-Lipschitz map $T: K\to K$ without fixed points.

Approximation properties and Schauder decompositions in Lipschitz-free spaces

We prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. We also show that the Lipschitz-free spaces over $\ell_1^N$ or $\ell_1$ have monotone finite-dimensional Schauder decompositions.

从 Schur $p$-性质与紧归约视角看 $0<p<1$ 的 Lipschitz free $p$-空间

Lipschitz free $p$-spaces for $0<p<1$ in the light of the Schur $p$-property and the compact reduction

本文引入 Schur $p$-性质及强 Schur $p$-性质，并利用适配的紧归约原理证明离散度量空间上的 Lipschitz free $p$-空间具有逼近性质，肯定回答了已有公开问题。

Explicit Formula for Preimages of Relaxed One-Sided Lipschitz Mappings with Negative Lipschitz Constants

This paper addresses Lipschitzian stability issues that play an important role in both theoretical and numerical aspects of variational analysis, optimization, and their applications. We particularly concentrate on the so-called relaxed one-sided Lipschitz property of set-valued mappings with negative Lipschitz constants. This property has been much less investigated than more conventional Lipschitzian behavior while being well recognized in a variety of applications. Recent work has revealed that set-valued mappings satisfying the relaxed one-sided Lipschitz condition with negative Lipschitz …

Lipschitz自由空间中的球覆盖性质

The Ball-Covering Property in Lipschitz-Free Spaces

本文研究Lipschitz自由空间的球覆盖性质，建立扩展判据证明某些空间不具该性质，构造反例显示定量性质非遗传，并证明小扰动下的稳定性。

Emerging notions of norm attainment for Lipschitz maps between Banach spaces

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in $\Lip(X,Y)$. For instance, we characterize a Banach space $Y$ with the Radon-Nikodým property in terms of the denseness of norm attaining Lipschitz maps with values in $Y$. Further, we introduce a property called the local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps, which extends the…

Approximation of Lipschitz functions by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces

This note corrects a gap and improves results in an earlier paper by the first named author. More precisely, it is shown that on weakly compactly generated Banach spaces X which admit a C^{p} smooth norm, one can uniformly approximate uniformly continuous functions f:X->R by Lipschitz, C^{p} smooth functions. Moreover, there is a constant C>1 so that any L-Lipschitz function f:X->R can be uniformly approximated by CL-Lipschitz, C^{p} smooth functions. This provides a `Lipschitz version' of the classical approximation results of Godefroy, Troyanski, Whitfield and Zizler.

Approximation by Lipschitz, analytic maps on certain Banach spaces

We show that on separable Banach spaces admitting a separating polynomial, any uniformly continuous, bounded, real-valued function can be uniformly approximated by Lipschitz, analytic maps on bounded sets.

Approximation by Lipschitz functions

On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular, on bounded symmetric domains in complex Euclidean space.

Approximation of Lipschitz functions by $Δ$-convex functions in Banach spaces

In this paper we give some results about the approximation of a Lipschitz function on a Banach space by means of $Δ$-convex functions. In particular, we prove that the density of $Δ$-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of $Δ$-convex functions.

A Universal Lipschitz Extension Property of Gromov Hyperbolic Spaces

A metric space has the universal Lipschitz extension property if for each subspace S embedded quasi-isometrically into an arbitrary metric space M there exists a continuous linear extension of Banach-valued Lipschitz functions on S to those on all of M. We show that the finite direct sum of Gromov hyperbolic spaces of bounded geometry is universal in the sense of this definition.

Banach空间上具有最大导数的Lipschitz映射的逼近

Approximations of Lipschitz maps with maximal derivatives on Banach spaces

本文研究Banach空间上Lipschitz映射的两类逼近，用强范数可达和最大导数可达映射刻画Radon-Nikodým性质，并证明实直线上Lipschitz泛函可被最大仿射函数局部逼近但无法一致逼近。

关于不可分离 Lipschitz-free 空间的一个注记

A note on nonseparable Lipschitz-free spaces

本文研究 Lipschitz-free 空间中经典 Banach 空间性质与可分离性的等价性，指出部分性质下等价失效、相关 weak* 序列紧球问题在 ZFC 中不可判定，并构造缺失 Radon–Nikodým 性质的反例。

Lipschitz slices versus linear slices in Banach spaces

The aim of this note is study the topology generated by Lipschitz slices in the unit sphere of a Banach space. We prove that the above topology agrees with the weak topology in the unit sphere and, as a consequence, we obtain Lipschitz characterizations of classical linear topics in Banach spaces, as Radon-Nikodym property, convex point of continuity property and strong regularity, which shows that the above classical linear properties only depend on the natural uniformity in the Banach space given by the metric and the linear structure.

On diagonalizable operators in Minkowski spaces with the Lipschitz property

A real semi-inner-product space is a real vector space $\M$ equipped with a function $[.,.] : \M \times \M \to \Re$ which is linear in its first variable, strictly positive and satisfies the Schwartz inequality. It is well-known that the function $||x|| = \sqrt{[x,x]}$ defines a norm on $\M$. and vica versa, for every norm on $X$ there is a semi-inner-product satisfying this equality. A linear operator $A$ on $\M$ is called \emph{adjoint abelian with respect to $[.,.]$}, if it satisfies $[Ax,y]=[x,Ay]$ for every $x,y \in \M$. The aim of this paper is to characterize the diagonalizable adjoint …

Compact reduction in Lipschitz free spaces

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal F(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal F(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

Real analytic approximations which almost preserve Lipschitz constants of functions defined on the Hilbert space

Let $X$ be a separable real Hilbert space. We show that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and for every $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ and $\textrm{Lip}(g)\leq \textrm{Lip}(f)+ε$.

Lipschitz自由空间中的非粗糙范数

Nonrough norms in Lipschitz free spaces

本文利用底层度量空间的新几何性质刻画了Lipschitz自由空间F(M)的非粗糙范数。

Schur $p$-性质对Lipschitz-free $p$-空间的结构性后果

Structural consequences of the Schur $p$-property for Lipschitz-free $p$-spaces

本文证明Lipschitz-free $p$-空间具有Schur $p$-性质，解决多个开放问题，并证实2009年关于$p$-Lipschitz提升性质的预测，揭示非局部凸空间线性与Lipschitz结构的差异。

Weak Compactness and Fixed Point Property for Affine Bi-Lipschitz Maps

In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-$(s)$ subsequence $(x_n)$ which admits an equivalent convex basic sequence. This fact is used to characterize weak-compactness of bounded, closed convex sets in terms of the generic fixed point property ($\mathcal{G}$-$FPP$) for the class of affine bi-Lipschitz maps. This result generalizes a theorem by Benavides, Japón Pineda and Prus previously proved for the class of continuous maps. We also introduce a relaxation of this notion …

Lipschitz Free Spaces over Locally Compact Metric Spaces

We prove that the Lipschitz free space over a certain type of discrete metric space has the Radon-Nikodým property. We also show that the Lipschitz free space over a complete, locally compact metric space has the Schur or approximation property whenever the Lipschitz free space over each compact subset also has this property.

Approximation and Schur properties for Lipschitz free spaces over compact metric spaces

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metr…

Corrigendum to "Approximation by C^{p}-smooth, Lipschitz functions on Banach spaces" [J. Math. Anal. Appl., 315 (2006), 599-605]

In this erratum, we recover the results from an earlier paper of the author's which contained a gap. Specifically, we prove that if X is a Banach space with an unconditional basis and admits a C^{p}-smooth, Lipschitz bump function, and Y is a convex subset of X, then any uniformly continuous function f: Y->R can be uniformly approximated by Lipschitz, C^{p}-smooth functions K:X->R. Also, if Z is any Banach space and f:X->Z is L-Lipschitz, then the approximates K:X->Z can be chosen CL-Lipschitz and C^{p}-smooth, for some constant C depending only on X.

Approximation by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces

The following result was announced in the earlier version(s) of this paper: On weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded, real-valued functions by Lipschitz, C^{p} smooth functions. Unfortunately, there is a gap in the proof which renders the proof in its present form incorrect

Sobolev mappings: Lipschitz density is not a bi-Lipschitz invariant of the target

We study a question of density of Lipschitz mappings in the Sobolev class of mappings from a closed manifold into a singular space. The main result of the paper shows that if we change the metric in the target space to a bi-Lipschitz equivalent one, than the property of the density of Lipschitz mappings may be lost.

关于 $\mathbb{R}^n$ 上 Lipschitz 自由空间的 Dunford-Pettis 性质

The Dunford-Pettis property for Lipschitz-free spaces over $\mathbb{R}^n$

本文证明了 $\mathbb{R}^n$ 上的 Lipschitz 自由空间具有 Dunford-Pettis 性质，并由此得出其子集上的空间不与缺乏该性质的可分 Banach 空间同构，且 $\mathcal{F}(\ell_p)$ 与特定直和不同构。

Weakly compact composition operators on spaces of Lipschitz functions

Let $X$ be a pointed compact metric space. Assuming that $\mathrm{lip}_0(X)$ has the uniform separation property, we prove that every weakly compact composition operator on spaces of Lipschitz functions $\mathrm{Lip}_0(X)$ and $\mathrm{lip}_0(X)$ is compact.

On the composition ideals of Lipschitz mappings

We study in this paper some property of Lipschitz mappings which admit factorization through an operator ideal. We try to construct Lipschitz cross-norms from known tensor norms in order to represent certain classes of Lipschitz mappings. Inspired by the definition of p-summing linear operators we introduce a new concpet in the the category of Lipschitz mappings that is called strictly Lipschitz p-summing.

关于$c_0$的性质(H)以及Gromov和Johnson的球面问题

On Property (H) of c_0 and Lipschitz Problems of Gromov and Johnson

本文证明零序列Banach空间$c_0$不具有有理性质(H)，从而解决了Kasparov-Yu问题的一个关键部分，并同时解决了Gromov和Johnson提出的一个相关问题。

Approximation properties in Lipschitz-free spaces over groups

We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group $G$ equipped with an arbitrary compatible left-invariant metric $d$, the Lipschitz-free space over $G$, $\mathcal{F}(G,d)$, satisfies the metric approximation property. We show also that, given a finitely generated group $G$, with its word metric $d$, from a class of groups admitting a certain special type of combing, which includes all h…

均匀离散度量空间上的Lipschitz-free空间具有3-Schur性质

Lipschitz-free spaces over uniformly discrete metric spaces are 3-Schur

本文证明了任何均匀离散度量空间上的Lipschitz-free空间具有3-Schur性质，通过分析空间结构与性质的关系，揭示了该类空间的特征。

The Lipschitz-free space over 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.

Sobolev-to-Lipschitz Property on QCD-spaces and Applications

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature dimension condition recently introduced in E. Milman, The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds, Comm. Pure Appl. Math. (to appear, arXiv:1908.01513v5). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat sem…

Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon Nikodym property

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.

Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces

Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.

Lipschitz函数空间中直径二性质与其弱星对应性质的分离

Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions

该研究针对实值Lipschitz函数Banach空间的公开问题，证明直径二性质与弱星对应性质的差异，构造相关空间示例，引入新充分条件并明确已知示例状态，推进了相关空间几何性质的研究。

Lipschitz-free 空间与 Bossard 的缩减论证

Lipschitz-free spaces and Bossard's reduction argument

本文通过描述集合论框架，利用 Bossard 的缩减论证，研究了 Lipschitz-free 空间与 Banach 空间之间的普遍性及逼近性质。

强可数维空间上的无Lipschitz空间与逼近性质

Lipschitz-free spaces over strongly countable-dimensional spaces and approximation properties

本文证明了对于紧致、可度量且强可数维的拓扑空间$T$，在一致收敛拓扑下，使得无Lipschitz空间具有度量逼近性质的度量集合在所有相容度量中是剩余集。

Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete

Let $M$ be a compact subset of a superreflexive Banach space. We prove that the Lipschitz-free space $\mathcal{F}(M)$, the predual of the Banach space of Lipschitz functions on $M$, has the Pełczyński's property ($V^\ast$). As a consequence, the Lipschitz-free space $\mathcal{F}(M)$ is weakly sequentially complete.