A. Brudnyi, Yu. Brudnyi
If a metric subspace $M^{o}$ of an arbitrary metric space $M$ carries a doubling measure $μ$, then there is a simultaneous linear extension of all Lipschitz functions on $M^{o}$ ranged in a Banach space to those on $M$. Moreover, the norm of this linear operator is controlled by logarithm of the doubling constant of $μ$.
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We show that the class of subspaces of c_0 is stable under Lipschitz isomorphisms. The main corollary is that any Banach space which is Lipschitz isomorphic to c_0 is linearly isomorphic to c_0.
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 …
We compute the Lipschitz-free spaces of subsets of the real line and characterize subsets of metric trees by the fact that their Lipschitz-free space is isometric to a subspace of $L_1$.
The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere
本文证明了球面Lipschitz函数空间上连续对偶平移不变赋值可唯一分解为0、1、2次齐次赋值之和，并否定了3次及以上非平凡赋值的可能性，同时刻画了相应空间的稠密子空间。
Invariant pointwise closed subspaces of Lipschitz spaces and their preduals
本文研究图上Lipschitz函数空间的不变逐点闭子空间及其预对偶空间，证明ℤᵈ上Lipschitz空间的平移不变逐点闭子空间要么有限维要么非可分，对应预对偶空间要么有限维要么非自反。
In 2010, Olson \& Robinson [Transactions of the American Mathematical Society, 362(1), 145-168] introduced the notion of an almost homogeneous metric space and showed that if $X$ is a subset of a Hilbert space such that $X-X$ is almost homogeneous, then $X$ admits almost bi--Lipschitz embeddings into Euclidean spaces. In this paper, we extend this result and we show that if $X$ is a subset of a Banach space such that $X-X$ is almost homogeneous at the origin, then $X$ can be embedded in a Euclidean space in an almost bi--Lipschitz way.
Invariant subspaces for free linearizations of Lipschitz maps
该研究探讨Lipschitz自由空间上由Lipschitz自映射诱导算子的不变子空间问题，证明了特定条件下常规自由线性化$\boldsymbol{\textit{f}}$及算子$T_{f,e}$存在非平凡不变子空间，并给出线性动力学相关推论。
For a domain bounded by homogeneous subsets of a Lipschitz graph, we show the Corona Theorem is affirmative.
We prove a Rademacher-type theorem for Lipschitz mappings from a subset of a Carnot group to a Banach homogeneous group, equipped with a suitably weakened Radon-Nikodym property. We provide a metric area formula that applies to these mappings and more generally to all almost everywhere metrically differentiable Lipschitz mappings defined on a Carnot group.
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.
We construct a complete metric space $M$ of cardinality continuum such that every non-singleton closed separable subset of $M$ fails to be a Lipschitz retract of $M$. This provides a metric analogue to the various classical and recent examples of Banach spaces failing to have linearly complemented subspaces of prescribed smaller density character.
A Proper Closed Subspace of the Lipschitz Dual Containing the Linear Dual
本文研究了Lipschitz自由空间中一个恰当的闭子空间，证明其作为偶空间的性质，并展示了该空间作为Banach代数的结构优势。
Hyperbolicity and uniformly Lipschitz affine actions on subspaces of $L^1$
本文利用 Mineyev 的 $\mathbb{Q}$-双组合和 Balasubramanya 的拟树作用刻画，证明了双曲群和无端双曲群在 $L^1$ 空间子空间上分别具有真一致 Lipschitz 仿射作用和无界轨道的一致 Lipschitz 仿射作用。
In this paper we extend classical Titchmarsh theorems on the Fourier transform of Hölder-Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for $L^2$-Hölder-Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini-Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients.
We prove a version of the Ando-Choi-Effros lifting theorem respecting subspaces, which in turn relies on Oja's principle of local reflexivity respecting subspaces. To achieve this, we first develop a theory of pairs of $M$-ideals. As a first consequence we get a version respecting subspaces of the Michael-Pełczyński extension theorem. Other applications are related to linear and Lipschitz bounded approximation properties for a pair consisting of a Banach space and a subspace. We show that in the separable case, the BAP for such a pair is equivalent to the simultaneous splitting of an associate…
Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$
本文研究了几乎不变的条件负定核，并将其应用于刻画在 $L^1$ 空间子空间上具有适当均匀利普希茨作用的群类。
Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces $Y$, as well as the inverse question, that is, the possible sizes of the metric space $M$ given that such a subspace $Y$ exists. We also show that if the metric space $M$ is $σ$-precompact, then th…
In this paper we study the maximal subspaces of continuous n-homogeneous polynomials on complex and real non separable Banach spaces. In the real case we will prove that if P is a 2-homogeneous polynomial and if there exist a k-dimensional P-maximal subspace then every P-maximal subspace is k-dimensional.
For a metric space $(K,d)$ the Banach space $\Lip(K)$ consists of all scalar-valued bounded Lipschitz functions on $K$ with the norm $\|f\|_{L}=\max(\|f\|_{\infty},L(f))$, where $L(f)$ is the Lipschitz constant of $f$. The closed subspace $\lip(K)$ of $\Lip(K)$ contains all elements of $\Lip(K)$ satisfying the $\lip$-condition $\lim_{0<d(x,y)\to 0}|f(x)-f(y)|/d(x,y)=0$. For $K=([0,1],| {\cdot} |^α)$, $0<α<1$, we prove that $\lip(K)$ is a proper $M$-ideal in a certain subspace of $\Lip(K)$ containing a copy of $\ell^{\infty}$.
In article is constructed a wide couple of pairwice non-isomorphic separable superreflexive Banach spaces E that are subspace homogeneous. Their conjugates are quotient homogeneous. None of this couple neither isomorphic to its Cartesian square nor has the approximation property. At the same time, any such E is isomorphic to E+ E+ W for some Banach space W and, hence, solves the Schroeder - Bernstein problem.
For graded Hilbert spaces $H$ and shift-like commuting tuples $T \in B(H)^n$, we show that each homogeneous joint invariant subspace $M$ of $T$ has finite index and is generated by its wandering subspace. Under suitable conditions on the grading $(H_k)_{k\geq 0}$ of $H$ the algebraic direct sum $\tilde{M} = \oplus_{k\geq 0} M \cap H_k$ becomes a finitely generated module over the polynomial ring $\mathbb C[z]$. We show that the wandering subspace $W_T(M)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_T(M)$ forms a minimal set of generators for the $\mathbb C[z]$-module $\…
We prove that if $M$ is an infinite complete metric space then the set of strongly norm-attaining Lipschitz functions $\SA(M)$ contains a linear subspace isomorphic to $c_0$. This solves an open question posed by V. Kadets and O. Roldán.
We establish the sharp rate of continuity of extensions of $\mathbb{R}^m$-valued $1$-Lipschitz maps from a subset $A$ of $\mathbb{R}^n$ to a $1$-Lipschitz maps on $\mathbb{R}^n$. We consider several cases when there exists a $1$-Lipschitz extension with preserved uniform distance to a given $1$-Lipschitz map. We prove that if $m>1$ then a given map is $1$-Lipschitz and affine if and only if such distance preserving extension exists for any $1$-Lipschitz map defined on any subset of $\mathbb{R}^n$. This shows a striking difference from the case $m=1$, where any $1$-Lipschitz function has such p…
Main results of the paper: (1) For any finite metric space $M$ the Lipschitz free space on $M$ contains a large well-complemented subspace which is close to $\ell_1^n$. (2) Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of graphs which are not necessarily graph automorphisms; (b) In the case of such…
We introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of any dual almost square Banach space. We also take advantage of this notion to provide some criterion of non-duality of some subspaces of scalar as well as vector valued Lipschitz functions.
We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius attaining functions in order to show that the denseness fails for an arbitrary Banach space, and study a Lipschitz version of Daugavet centers. Furthermore, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe-Bochner spaces, and Banach spaces which contain a dense union of increasing family of …
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…
Polynomials as Lipschitz maps on the Veronese cone
该研究构造了度量空间$\mathbb{V}_X^d$，证明其上度量与嵌入张量空间范数无关，进一步得出齐次多项式的Lipschitz $q$-可和性等价于其对应Lipschitz映射的该性质，推广了线性算子的已知结论。
Let $\mathcal{P}_{K} (^{n}E; F)$ (resp. $\mathcal{P}_{w} (^{n}E; F)$) the subspace of all $P\in \mathcal{P}(^{n}E; F)$ which are compact (resp. weakly continuous on bounded sets). We show that if $\mathcal{P}_{K} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{K} (^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$. Likewise we show that if $\mathcal{P}_{w} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{w}(^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$.
Let $E$ be a finite-dimensional normed space and $Ω$ a nonempty convex open set in $E$. We show that the Lipschitz-free space of $Ω$ is canonically isometric to the quotient of $L^1(Ω,E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$.
We initiate the study of the forward and backward shifts on the Lipschitz space of a tree, $\mathcal L$, and on the little Lipshitz space of a tree, ${\mathcal L}_0$. We determine that the forward shift is bounded both on $\mathcal L$ and on ${\mathcal L}_0$ and, when the tree is leafless, it is an isometry; we also calculate its spectrum. For the backward shift, we determine when it is bounded on $\mathcal L$ and on ${\mathcal L}_0$, we find the norm when the tree is homogeneous, we calculate the spectrum for the case when the tree is homogeneous, and we determine, for a general tree, when it…
Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes E of a pointed metric space and a Banach space E as a certain linear subspace of the algebraic dual of Lipo(X,E^*). We prove that <Lipo(X,E^*),X\boxtimes E> forms a dual pair. We prove that X\boxtimes E is linearly isomorphic to the linear space of all finite-rank continuous linear operators from (X^#,T) into E, where X^# denotes the space Lipo(X,K) and T is the topology of pointwise convergence of X^#. The concept of Lipschitz tensor produc…
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。
Group Actions, Fixed Points and Orbit Quotients in Lipschitz Spaces
本文通过相似性群作用的归一化等距表示，推广轨道商方法，得到无可均性假设的保范线性化，并证明正齐次Lipschitz函数空间为1-补子空间，对可均群给出对偶空间中的等距实现。
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.
We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and $C$-rich subspaces have Lipschitz numerical index 1. Moreover, using the G$\hat{a}$teaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Fi…
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…
The Aron--Rueda zero-subspace problem
该研究确定了阿龙-鲁埃达零子空间问题中复齐次多项式及有界次数多项式的精确有限维阈值，两类阈值不同，分别通过特韦列夫定理、德巴尔-马内维尔定理及陈类等方法完成推导与证明。
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…
The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of the completion of a generalized b-metric space and some fixed point results. The behavior of Lipschitz functions on b-metric spaces of homogeneous type, as well as of Lipschitz functions defined on, or with values in quasi-Banach spaces, is studied.
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,…
The numerical index of $2$-dimensional Lipschitz-free spaces
本研究给出二维Lipschitz-free空间数值指数的显式公式及达极值算子的构造，证明其数值指数可取[1/2,1]内任意值，且该范围可通过高维Lipschitz-free空间的二维子空间实现。
We review the current state of the homogeneous Banach space problem. We then formulate several questions which arise naturally from this problem, some of which seem to be fundamental but new. We give many examples defining the bounds on the problem. We end with a simple construction showing that every infinite dimensional Banach space contains a subspace on which weak properties have become stable (under passing to further subspaces). Implications of this construction are considered.
Homogeneous Sobolev and Besov spaces on special Lipschitz domains and their traces
本文研究了特殊Lipschitz区域上齐次Sobolev和Besov空间的迹算子，证明了在精确正则性范围 $s \in (\frac{1}{p}, 1 + \frac{1}{p})$ 内迹定理成立，并处理了端点情况，填补了齐次函数空间研究的空白。
We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study $N$-body type Hamiltonians with interactions. More precisely, let $Y$ be a linear subspace of a finite dimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$ that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form $H = - Δ+ \sum_{Y \in S} v_Y$, where the subspaces $Y$ belong to some given family S of subspaces. We prove results on the spectral theory of the Hamiltonian when $S$ is any family of subspaces and exten…
In this paper, we provide an infinite metric space $M$ such that the set $\mbox{SNA}(M)$ of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to $c_0$. This answers a question posed by Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete. On the other hand, we prove that $\mbox{SNA}(M)$ contains an isometric copy of $c_0$ whenever $M$ is a metric space which is not uniformly discrete. In particular, the latter holds true for infinite compact metric spaces while it does not for proper metric spaces. Some positive results in…
In any quasi-metric space of homogeneous type, Auscher and Hytönen recently gave a construction of orthonormal wavelets with Hölder-continuity exponent $η>0$. However, even in a metric space, their exponent is in general quite small. In this paper, we show that the Hölder-exponent can be taken arbitrarily close to 1 in a metric space. We do so by revisiting and improving the underlying construction of random dyadic cubes, which also has other applications.
We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and…
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 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。
On complementation and $L$-embedding of Lipschitz-free spaces in their biduals
本文研究Lipschitz自由空间的性质，证明具有局部有限H¹测度的真测地度量空间对应的Lipschitz自由空间在其二次对偶中是L-嵌入的，还推广相关结果得到有限维赋范空间对应的Lipschitz自由空间的补性结论。
Isometric copies of $\ell_\infty^n$ in Lipschitz-free spaces over finite metric spaces
针对每个自然数n，构造有限度量空间M_n，使其Lipschitz-free空间包含ℓ_∞^n的等距拷贝，从而解决Khan等人提出的问题，并刻画可嵌入的有限维空间为多面体空间。
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…
An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces
本文针对有限维空间中具有内蕴距离的区域，给出了Lipschitz-free空间的等距表示，并推广了已有结果。
We shall consider the truncated singular integral operators T_{μ, K}^εf(x)=\int_{\mathbb{R}^{n}\setminus B(x,ε)}K(x-y)f(y)dμy and related maximal operators $T_{μ,K}^{\ast}f(x)=\underset{ε>0}{\sup}| T_{μ,K}^εf(x)|$. We shall prove for a large class of kernels $K$ and measures $μ$ and $ν$ that if $μ$ and $ν$ are separated by a Lipschitz graph, then $T_{ν,K}^{\ast}:L^p(ν)\to L^p(μ)$ is bounded for $1<p<\infty$. We shall also show that the truncated operators $T_{μ, K}^ε$ converge weakly in some dense subspaces of $L^2(μ)$ under mild assumptions for the measures and the kernels.
Since the pioneering works by Aronson & Bénilan [C. R. Acad. Sci. Paris Sér., 1979] and Bénilan & Crandall [Johns Hopkins Univ. Press, 1981], it is well-known that first-order evolution problems governed by a nonlinear but homogeneous operator admit the smoothing effect that every corresponding mild solution is Lipschitz continuous at every positive time. Moreover, if the underlying Banach space has the Radon-Nikodým property, then these mild solution is a.e. differentiable, and the time-derivative satisfies global and point-wise bounds. In this paper, we show that these results remain true if…
In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of $\ell_1$. This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space $K$ such that $F(K)$ is not isomorphic to a subspace of $L_1$ and we show that whenever $M$ is a subset of $R^n$, then $F(M)$ is weakly sequentially complete; in particular, $c_0$ does not embed into $F(M)$.
Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces
本文研究了Banach空间上线性余切的Oseledets子空间，在可逆与不可逆情形下证明了其Hölder连续性，推广了已有结果。
Zero sets of homogeneous polynomials containing infinite dimensional spaces
本文研究了无限维线性空间上齐次多项式零点集的性质，证明了若多项式在有限维子空间为零，则存在包含该子空间的无限维子空间使其为零，并推广了复情形的经典结果。
Linearization of holomorphic Lipschitz functions
本文研究复Banach空间开单位球上的全纯Lipschitz函数空间，借助Dixmier-Ng定理明确其预对偶的线性化性质，得到子空间结构、逼近性质及泛函延拓刻画等结果。
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