一类巴拿赫空间上解析映射的Lipschitz逼近

Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces

In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for $1\le q<p$, a $q$-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a $p$-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga.

Some properties of coarse Lipschitz maps between Banach spaces

We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a result of G. Godefroy on Lipschitz equivalences. This leads us to include the non separable versions of classical results on the stability of the existence of asymptotically uniformly smooth norms under Lipschitz or coarse Lipschitz equivalences.

Octahedrality in Lipschitz free Banach spaces

The aim of this note is to study octahedrality in vector valued Lipschitz-free Banach spaces on a metric space under topological hypotheses on it. As a consequence, we get that the space of Lipschitz functions on a metric space valued in a dual Banach space satisfies the weak-star strong diameter two property, under natural topological hipothesess on the metric space. Also, we show an example proving that these hypotheses are optimal.

经典巴拿赫空间中Lipschitz收缩常数的改进界

Improved Bounds For The Lipschitz Retraction Constant In Classical Banach Spaces

本文改进了经典无穷维巴拿赫空间中单位球到单位球面的最小Lipschitz收缩常数的上下界，针对多个经典空间。

弱紧生成巴拿赫空间上Lipschitz，C^{p}光滑函数对Lipschitz函数的逼近

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

本说明纠正了第一命名作者的早期论文中的一个差距并改进了结果。更精确地说，证明了在弱紧生成的巴拿赫空间X上，若X具有C^{p}光滑范数，则可以用Lipschitz，C^{p}光滑函数一致逼近一致连续函数f：X->R.此外，存在一个常数C>1，使得任意L-Lipschitz函数f：X->R都可以用CL-Lipschitz，C^{p}光滑函数一致逼近.这提供了一个'Lipschitz版本'的经典近似结果的Goaly，Troyanski，Whitfield和Zizler。

The approximation property and Lipschitz mappings on Banach spaces

We present an overview to the approximation property, paying especial attention to the recent results relating the approximation property to ideals of linear operators and Lipschitz ideals. We complete the paper with some new results on approximation of Lipschitz mappings and their relation to linear operator ideals.

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.

Galois connection between Lipschitz and linear operator ideals and minimal Lipschitz operator ideals

We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allow us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. Also we characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.

Schauder bases in Lipschitz free spaces over nets in Banach spaces

In the present note we give two explicit constructions (based on a retractional argument) of a Schauder basis for the Lipschitz free space $\mathcal{F}(N)$, over certain uniformly discrete metric spaces $N$. The first one applies to every net $N$ in a finite dimensional Banach space, leading to the basis constant independent of the dimension. The second one applies to grids in Banach spaces with an FDD. As a corollary, we obtain a retractional Schauder basis for the Lipschitz free space $\mathcal{F}(N)$ over a net $N$ in every Banach space $X$ with a Schauder basis containing a copy of $c_0$, …

Lipschitz retractions and complementation properties of Banach spaces

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space $\mathcal{F}(X)$ is a Plichko space whenever $X$ is a Plichko Banach space. Our main results include two examples of metric spaces. The first one $M$ contains two points $\{0,…

Asymptotic structure and coarse Lipschitz geometry of Banach spaces

In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly smoothness and convexity, and several distinct Banach-Saks-like properties. Among other results, we characterize the Banach spaces which are either coarsely or uniformly homeomorphic to $T^{p_1}\oplus \ldots \oplus T^{p_n}$, where each $T^{p_j}$ denotes the $p_j$-convexification of the Tsirelson space, for $p_1,\ldots,p_n\in (1,\ldots, \infty)$, and $2\not\in\{p_1,\ldots ,p_n\}$. We obtain applications to the coarse Lipschitz geometry of t…

关于Banach空间单位球上A-紧全纯Lipschitz映射

A-compact holomorphic Lipschitz mappings on the unit ball of a Banach space

本文研究Banach空间单位球上A-紧全纯Lipschitz映射的结构与性质，探讨其作为Banach全纯Lipschitz理想的关系。

关于巴拿赫空间中有限余维的Lipschitz曲面和d.c.曲面

On Lipschitz and d.c. surfaces of finite codimension in a Banach space

研究了巴拿赫空间中有限余维Lipschitz曲面和d.c.曲面的性质以及生成$σ$ -理想的性质.这些$σ$ -理想自然地出现在微分论和抽象的逼近论中。利用这些性质，改进了M. Heisler给出了D.关于凸函数的奇点。

度量空间与Banach空间之间双Lipschitz算子的理想构造

Building ideals of two-Lipschitz operators between mertic and Banach spaces

该研究针对度量空间与Banach空间之间的双Lipschitz算子理想问题，刻画其单射包与对偶理想定义，提出Lipschitz化和因子分解两种构造方法，证明新理想的闭性、单射性与对称性依赖原理想的对应性质并给出示例。

“巴拿赫空间上C^{p}-光滑Lipschitz函数的逼近“[J. Math. Anal.应用程序、315（2006），599-605]

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

在这个勘误表中，我们恢复了作者早期论文中的结果，其中包含了一个缺口。具体地说，我们证明了：如果X是一个具有无条件基的巴拿赫空间，且X中存在一个C^{p}-光滑Lipschitz冲击函数，且Y是X的凸子集，则任何一致连续函数f：Y->R都可以用Lipschitz，C^{p}-光滑函数K：X->R一致逼近.同样，如果Z是任意巴拿赫空间，且f：X->Z是L-Lipschitz，则近似K：X->Z可以是CL-Lipschitz和C^{p}-光滑的，对于某个常数C只依赖于X。

Extension of Lipschitz-type operators on Banach function spaces

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced.%more appropriate. Thus, we adapt the classical definition of Lipschitz map to the context of spaces of integrable functions by introducing such elements. We analyze Lipschitz type inequalities in two fundamental cases. The first concerns a.e. pointwise inequalities, while the second considers dominations involving …

某些 Banach 空间中的 Lipschitz 数值指数与线性数值指数

Lipschitz vs Linear Numerical Index in certain Banach spaces

针对实 Banach 空间中 Lipschitz 与线性数值指数是否一致的问题，本文利用 Gaussian 卷积微分和不变均值两种线性化技术，证明在可分或对偶空间中二者相等。

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…

Lipschitz，C^{p}光滑函数在弱紧生成巴拿赫空间上的逼近

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

本文的前几个版本中已经宣布了以下结果：在弱紧生成的巴拿赫空间中，如果存在Lipschitz，C^{p}光滑冲击函数，则可以用Lipschitz，C^{p}光滑函数一致逼近一致连续、有界、实值函数.不幸的是，证据中有一个漏洞，使目前形式的证据不正确

逐点半Lipschitz函数与Banach-Stone定理

Pointwise semi-Lipschitz functions and Banach-Stone theorems

本文研究非对称空间上逐点半Lipschitz函数的基本性质及部分对称性的影响，并利用两个函数代数建立两个Banach-Stone型结果，其核心是一个适用于抽象函数空间的灵活拓扑型一般定理。

Lipschitz $p$ -凸映射和$q$ -凹映射

Lipschitz $p$-convex and $q$-concave maps

$p$ -凸性和$q$ -凸性的概念最为人所知，因为它们作为研究巴拿赫格同构性质的工具的重要性，但它们也在涉及巴拿赫空间和巴拿赫格之间的线性映射的几个结果中发挥作用。在本文中，我们介绍Lipschitz版本的这些概念，处理度量空间和巴拿赫格之间的映射，并通过证明非线性版本的两个著名的因子分解定理通过$L_p$空间由于Maurey/Nikishin和Krivine。我们还证明了从度量空间到巴拿赫格的Lipschitz映射是Lipschitz $p$ -凸的当且仅当它的线性化是$p$ -凸的。此外，我们阐明了为什么有这样一个密切的关系之间的线性和非线性的概念，证明特征的Lipschitz $p$ -凸和Lipschitz $q$ -凹映射的因式分解通过$p$ -凸和$q$ -凹巴拿赫格，分别在雷诺和Tradacete的工作的精神。

一个具有绍德尔基的可分巴拿赫空间，它不是其二对偶空间的利普希茨收缩核

A Separable Banach Space with a Schauder Basis Which Is Not a Lipschitz Retract of Its Bidual

构造具有绍德尔基的可分巴拿赫空间\(X\)，证明\(B_X\)不是\(B_{X^{**}}\)的一致连续收缩核，得出\(X\)不是\(X^{**}\)的利普希茨收缩核的结论。

几乎双利普希茨嵌入与巴拿赫空间的真子集——M.I. Ostrovskii一个定理的推广

Almost bi-Lipschitz embeddings and proper subsets of a Banach space -- An extension of a theorem by M.I. Ostrovskii

该研究推广了M.I. Ostrovskii的定理，证明当无限维巴拿赫空间X粗有限可表示于Y的每个有限余维子空间时，X的任意真子集可几乎双利普希茨嵌入Y，拓展了原有局部有限子集情形的结论。

Surjection and inversion for locally Lipschitz maps between Banach spaces

We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these notions, we derive our results about existence and uniqueness of solution for nonlinear equations. In particular, …

可分Banach空间上的Lipschitz-free空间及其在二次对偶中的可补性

Lipschitz-free spaces over separable Banach spaces and their complementability in the bidual

研究可分Banach空间上的Lipschitz-free空间在二次对偶中的可补性，证明其可补当且仅当原空间同构于Hilbert空间，否定了Godefroy等人的相关问题，还刻画了范数1投影的存在性并推广到凸子集。

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。

Lipschitz函数在希尔伯特空间及其它巴拿赫空间上的实解析逼近

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

设$X$是具有分离多项式的可分Banach空间.证明了存在$C\geq 1$（仅依赖于$X$）使得对于每个Lipschitz函数$f:X\rightarrow\mathbb{R}$和每个$ε>0$，存在一个Lipschitz实解析函数$g:X\rightarrow\mathbb{R}$使得$|f(x)-g(x)|\leq ε$和$\textrm{Lip}(g)\leq C\textrm{Lip}(f)$ .即使在$X$是希尔伯特空间的情况下，这个结果也是新的。此外，在希尔伯特的情况下，我们还表明，$C$可以被假定为任何大于1的数字。

Compact retractions and Schauder decompositions in Banach spaces

In our note we show the very close connection between the existence of a Finite Dimensional Decomposition (FDD for short) for a separable Banach space $X$ and the existence of a Lipschitz retraction of $X$ onto a small (in a certain precise sense) generating convex and compact subset $K$ of $X$. In one direction, if $X$ admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set $K$. On the other hand, we prove that if $X$ admits a small generating compact Lipschitz retract then $X$ has the $π$-property. We note that it is still unknown if the $π$-prop…

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…

Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.

在p-巴拿赫空间中的Lipschitz扩展，以及Lipschitz-free p-空间的规范嵌入

Lipschitz extensions into $p$-Banach spaces, and canonical embeddings of Lipschitz-free $p$-spaces for $0<p<1$

研究了p-度量空间的包含关系对Lipschitz-free p-空间的线性嵌入影响，证明了规范映射总是同胚嵌入，并探讨了Lipschitz-free p-空间的结构特性。

$p$-Banach 空间中 Lipschitz 曲线的可微性

Differentiability of Lipschitz curves in $p$-Banach spaces

本文刻画了拟Banach空间中每条Lipschitz曲线可微的充要条件为同构于具有Radon–Nikodým性质的Banach空间，并研究$\ell_p$（$0<p<1$）中解耦坐标构造下的几乎处处可微性。

Lipschitz映射理想的对偶性

Duality for ideals of Lipschitz maps

我们开发了一个系统的方法来研究理想的Lipschitz映射从度量空间到巴拿赫空间的对偶，灵感来自经典理论，涉及理想的运营商和张量规范的巴拿赫空间，通过使用Lipschitz张量积以前介绍了相同的作者。我们首先研究了从度量空间到对偶巴拿赫空间的Lipschitz映射空间，它可以正则地表示为赋Lipschitz交叉范数的Lipschitz张量积的对偶。我们表明，几个已知的例子理想的Lipschitz地图（Lipschitz地图，Lipschitz $p$ -求和地图和地图承认一个Lipschitz分解通过一个子集的$L_p$空间）承认这样的表示，更一般地说，我们刻画当一个空间的Lipschitz地图从度量空间的一个双巴拿赫空间是在规范的对偶与Lipschitz交叉规范。此外，我们给出了几乎等价于Lipschitz映射空间具有理想性质的Lipschitz交范数的条件。我们引入了一个概念的运营商是近似的这些理想之一的Lipschitz映射，并确定他们的张量积概念。最后，我们还证明了一个Lipschitz版本的表现定理的极大算子理想。这使我们能够将Lipschitz交叉范数与取值于一般巴拿赫空间而不仅仅是对偶空间的Lipschitz映射的理想联系起来。

一类不可分巴拿赫空间上函数的光滑延拓

Smooth extension of functions on a certain class of non-separable Banach spaces

让我们考虑一个Banach空间$X$的性质，每个实值Lipschitz函数$f$可以一致近似的Lipschitz，$C^1$ -光滑函数$g$与$\Lip(g)\le C \Lip(f)$（与$C$只取决于空间$X$）。这是一个Banach空间$X$双Lipschitz同胚的$c_0(Γ)$的一个子集的情况下，对于一些集$Γ$，使得同胚的坐标函数是$C^1$ -光滑的。然后，证明了对每个闭子空间$Y\subset X$和每个$C^1$ -光滑（Lipschitz）函数$f:Y\to\Real$，存在$f$到$X$的$C^1$ -光滑（Lipschitz）扩张.我们还研究了定义在$X$闭子集上的实值函数的$C^1$ -光滑扩张。

Projecting Lipschitz functions onto spaces of polynomials

The Banach space $\mathcal{P}({}^2X)$ of $2$-homogeneous polynomials on the Banach space $X$ can be naturally embedded in the Banach space ${{\rm Lip}_0}(B_X)$ of real-valued Lipschitz functions on $B_X$ that vanish at $0$. We investigate whether $\mathcal{P}({}^2X)$ is a complemented subspace of ${{\rm Lip}_0}(B_X)$. This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that $\mathcal{P}({}^1X)=X^*$ is complemented in ${{\rm Lip}_0}(B_X)$ for every Banach space $X$. Our main result asserts that $\mathcal{P}({}^2X)$ is not c…

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

We study several properties and applications of the ultrapower $M_{\mathcal U}$ of a metric space $M$. We prove that the Lipschitz-free space $\mathcal F(M_{\mathcal U})$ is finitely representable in $\mathcal F(M)$. 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 $\mathcal F(M)$ is finitely representable in $\mathcal F(X)$. We apply these results to the study of coty…

度量空间与巴拿赫空间之间的非线性算子理想（PART I）

Nonlinear Operator Ideals Between Metric Spaces and Banach Spaces (PART I)

本文给出了度量空间与巴拿赫空间之间的非线性算子理想理论的第一部分。本文在A. Pietsch我们提出了三种类型的非线性版本的运营商理想。我们介绍了几个非线性理想的例子和它们之间的关系。对于每个空间理想$\mathsf{A}$都可以由一个特殊的非线性理想生成，该理想由允许通过Banach空间$\mathbf{M}\in\mathsf{A}$的分解的Lipschitz算子组成。我们研究了非线性理想的乘积和乘积。我们致力于从给定的理想构造三种新的非线性理想。“new”是分别为每个$\mathfrak{A}$、$\textswab{A}^{L}$和$\textfrak{A}^{L}$定义非线性理想$\mathfrak{A}^{L}_{new}$、$\textswab{A}^{L}_{new}$和$\textfrak{A}^{L}_{new}$的规则，称为Lipschitz过程。考虑到一类稳定的目标，对于一个给定的Lipschitz程序，我们得到了具有特殊性质的非线性理想。提出了（强）$p-$ Banach非线性理想（$0<p<1$）的概念，证明了Lipschitz核算子、Lipschitz Hilbert算子、乘积和商的非线性理想都是强$r-$ Banach非线性理想（$0<r<1$）.

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.

Banach空间中Lipschitzian半压缩与α-半压缩半群的隐式迭代的弱收敛与强收敛

Weak and Strong Convergence of an Implicit Iteration for Hemi-Contractive Semigroups in Banach Spaces

针对Banach空间中Lipschitzian半压缩和α-半压缩半群，提出隐式迭代格式逼近公共不动点，在满足Opial条件的均匀凸Banach空间中建立弱收敛，在一般Banach空间中获得强收敛，并推广到伪压缩和半压缩映射。

关于0<p<1时Lipschitz自由p-空间的Banach包络与对偶

On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$

本文研究0<p<1时Lipschitz自由p-空间的Banach包络，证明其Banach包络映射为单射，正面解决了Kalton提出的问题，且该性质对空间线性结构有重要应用，最直接的是对偶空间可分离空间的点。

Lipschitz-free Banach 空间的等距映射

Isometries of Lipschitz-free Banach spaces

本文研究一大类 Lipschitz-free 空间的线性等距映射与等距群，提出 Lipschitz-free 刚性度量空间概念，并证明其包含 3-连通图和非交换 Carnot 群等例子，且任意度量空间只需增加三点即可嵌入此类刚性空间。

关于巴拿赫空间间Lipschitz映射的Fréchet可微性

On Fréchet differentiability of Lipschitz maps between Banach spaces

一个著名的公开问题是：在具有可分对偶的巴拿赫空间X上，Lipschitz函数的可数集合是否有一个公共的Frechet可微点。我们证明了对于某些无限维X，答案是肯定的。以前，即使是由两个函数组成的集合，这也只对有限维X是已知的（尽管对于一个函数，答案在完全一般性上是肯定的）。通过在巴拿赫空间中引入一类新的空集（称为$Γ$-null sets），我们的目的得以实现，它的定义包含了范畴和测度的概念，并证明了所需的可微性关于它几乎处处成立，我们甚至得到了某些无限维巴拿赫空间之间Lipschitz函数的Fréchet导数的存在性;以前没有这样的结果。我们的主要结果表明，可分巴拿赫空间之间的Lipschitz映射是Fréchet可微的$Γ$-几乎处处，只要它是正则的Gateaux可微的$Γ$-几乎处处，并且Gateaux导数保持在一个范数可分空间内.很容易看出，X到具有Radon-Nikodym性质的空间的Lipschitz映射$Γ$-几乎处处是Gateaux可微的。此外，除了另一类可忽略集（即所谓的$σ$-多孔集）之外，Gateaux可微性还包含正则Gateaux可微性.因此，在每个$σ$-多孔集为$Γ$-空的空间中，问题的答案是肯定的。我们证明了这对具有$K$可数紧的$C(K)$、Tsirelson空间和$c_0$的所有子空间都成立，但对希尔伯特空间不成立.

Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

In this paper, we study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space $X,$ the Lipschitz algebras $Lip_α(X)$ and $\ell ip_α(X)$ are approximately biflat if and only if $X$ is finite, provided that $0<α<1$. We give an enough and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible for each $α>0.$ We also show that some triangular Banach algebras are not approximately biflat.

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。

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.

Interpolation of nonlinear positive or order preserving operators on Banach lattices

We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo-Peetre operators, and exact partially $K$-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple $(L^1,L^\infty )$ due to Bénilan and Crandall.

Banach 函数空间上的格 Lipschitz 叠加算子

Lattice Lipschitz superposition operators on Banach function spaces

本文刻画了 Banach 函数空间之间的格 Lipschitz 叠加算子，证明其良定义性等价于可由属于 Bochner 空间 $\mathcal M(X,Y)(μ, Lip_0(\mathbb R))$ 的强可测函数表示，并揭示了其与赋范张量积等映射类的关系。

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

A norm one element $x$ of a Banach space is a Daugavet-point (respectively, a $Δ$-point) if every slice of the unit ball (respectively, every slice of the unit ball containing $x$) contains an element, which is almost at distance 2 from $x$. We characterize Daugavet- and $Δ$-points in Lipschitz-free spaces. Furthermore, we construct a Lipschitz-free space with the Radon--Nikodým property and a Daugavet-point; this is the first known example of such a Banach space.

巴拿赫空间上的Lipschitz函数空间

Spaces Of Lipschitz Functions On Banach Spaces

R. C.詹姆斯如下：设$X$是Banach空间，$C \subseteq X$是范数有界闭凸集，使得每个线性泛函$x^* \in X^*$都在$C$上达到上确界，则$C$是弱紧集.实际上，这个结果比这个陈述强得多;事实上，这个证明可以用来获得其他令人惊讶的结果。例如，假设$X$是一个可分的Banach空间，而$S$是$X^*$的单位球的一个范数可分的子集，使得对于每个$x \in X$都存在$x^* \in S$，使得$x^*(x) = \|x\|$则$X^*$本身是范数可分的。如果我们称$S$为支持集，在这种情况下，相对于整个空间$X$，人们可以询问关于支持集的大小和结构的问题，支持集不仅相对于$X$本身，而且可能相对于$X$ @的某个其他子集。我们分析了一个特殊的情况下，这以及给出一些应用。

Porosity and Differentiability of Lipschitz Maps from Stratified Groups to Banach Homogeneous Groups

Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $σ$-porous set. At density points of $A$ we establish a pointwise characterization of differentiability in terms of directional derivatives. We use these new results to obtain an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property. As a consequence we …

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。

Banach 空间中的局部补全及其在自由构造下的保持性

Local complementation in Banach spaces and its preservation under free constructions

本文研究 Banach 空间中局部补全的推广概念，证明其与经典定义等价，并表明该性质在 Lipschitz 自由空间和自由 Banach 格等自由构造下得以保持。

线性算子的Lipschitz延拓

Lipschitz extensions of linear operators

本文证明线性算子可延拓至极大Banach算子理想当且仅当其可延拓至相关Lipschitz算子理想，并推论特殊Lipschitz分解蕴含线性分解。

将$c_0$粗李普希茨嵌入可分对偶巴拿赫空间

A Coarse-Lipschitz Embedding of $c_0$ into a Separable Dual Banach Space

该研究构造了$c_0$到可分对偶巴拿赫空间的粗李普希茨嵌入，证明最优失真为2，且利用Kalton环形分解实现了任意小的失真逼近。