On Lipschitz-free spaces over spheres of Banach spaces

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。

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.

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。

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

Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space, and properties of generated $σ$-ideals are studied. These $σ$-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.

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。

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.

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…

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

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

The notions of $p$-convexity and $q$-concavity are mostly known because of their importance as a tool in the study of isomorphic properties of Banach lattices, but they also play a role in several results involving linear maps between Banach spaces and Banach lattices. In this paper we introduce Lipschitz versions of these concepts, dealing with maps between metric spaces and Banach lattices, and start by proving nonlinear versions of two well-known factorization theorems through $L_p$ spaces due to Maurey/Nikishin and Krivine. We also show that a Lipschitz map from a metric space into a Banac…

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。

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.

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。

Duality for ideals of Lipschitz maps

We develop a systematic approach to the study of duality for ideals of Lipschitz maps from a metric space to a Banach space, inspired by the classical theory that relates ideals of operators and tensor norms for Banach spaces, by using the Lipschitz tensor products previously introduced by the same authors. We first study spaces of Lipschitz maps, from a metric space to a dual Banach space, that can be represented canonically as the dual of a Lipschitz tensor product endowed with a Lipschitz cross-norm. We show that several known examples of ideals of Lipschitz maps (Lipschitz maps, Lipschitz …

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

Let us consider a Banach space $X$ with the property that every real-valued Lipschitz function $f$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (with $C$ depending only on the space $X$). This is the case for a Banach space $X$ bi-Lipschitz homeomorphic to a subset of $c_0(Γ)$, for some set $Γ$, such that the coordinate functions of the homeomorphism are $C^1$-smooth. Then, we prove that for every closed subspace $Y\subset X$ and every $C^1$-smooth (Lipschitz) function $f:Y\to\Real$, there is a $C^1$-smooth (Lipschitz, respectively) extens…

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…

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

In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear versions of operator ideals. We introduce several examples of nonlinear ideals and the relationships between them. For every space ideal $\mathsf{A}$ can be generated by a special nonlinear ideal which consists of those Lipschitz operators admitting a factorization through a Banach space $\mathbf{M}\in\mathsf{A}$. We investigate products and quotients of nonlinea…

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。

On Fréchet differentiability of Lipschitz maps between Banach spaces

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called $Γ$-null sets), whose definition involves both the notions of category…

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.

Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces

We prove that an onto isometry between unit spheres of finite-dimensional polyhedral Banach spaces extends to a linear isometry of the corresponding spaces.

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.

On Uniform Homeomorphisms of the Unit Spheres of Certain Banach Lattices

We prove that if X is an infinite dimensional Banach lattice with a weak unit then there exists a probability space (Omega, Sigma,mu) so that the unit sphere S(L_1(Omega, Sigma, mu) is uniformly homeomorphic to the unit sphere S(X) if and only if X does not contain l_{infty}^n's uniformly.

Spaces Of Lipschitz Functions On Banach Spaces

A remarkable theorem of R. C. James is the following: suppose that $X$ is a Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such that every linear functional $x^* \in X^*$ attains its supremum on $C$; then $C$ is a weakly compact set. Actually, this result is significantly stronger than this statement; indeed, the proof can be used to obtain other surprising results. For example, suppose that $X$ is a separable Banach space and $S$ is a norm separable subset of the unit ball of $X^*$ such that for each $x \in X$ there exists $x^* \in S$ such that $x^*(x) = \|x\|$ then…

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。