Let $X$ be a Banach space and let $(ξ_j)_{j\ge 1}$ be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent:
(1). There exists a constant $K$ such that $$ \Bigl(\E\Big\|\sum_{j=1}^n ξ_j f(x_j)\Big\|^2\Bigr)^{\frac12} \leq K \n f\n_{\rm Lip} \Bigl(\E\Big\|\sum_{j=1}^n ξ_j x_j\Big\|^2\Bigr)^{\frac12} $$ for all Lipschitz functions $f:X\to X$ satisfying $f(0)=0$ and all finite sequences $x_1,...,x_n$ in $X$.
(2). $X$ is isomorphic to a Hilbert space.
We give a complete description of homomorphisms between two Banach algebras of Lipschitz functions with values in continuous functions. We also characterize the compactness of those homomorphisms.
Let \(X\) be a compact metric space and \(E\) be a Banach space. \(\lip (X, E)\) denotes the Banach space of all \(E\)-valued little Lipschitz functions on \(X\). We show that \(\lip (X, E)^{**}\) is isometrically isomorphic to Banach space of \(E^{**}\)-valued Lipschitz functions \(\Lip(X, E^{**})\) under several conditions. Moreover, we describe the isometric isomorphism from \(\lip (X, E)^{**}\) to \(\Lip (X, E^{**})\).
We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called {\bf Politics} - a generalization of a game called {\bf Tug of War} that has been used in \cite{PSSW09} to study real-valued AMLEs.
We analyse the strong connections between spaces of vector-valued Lipschitz functions and spaces of linear continuous operators. We apply these links to study duality, Schur properties and norm attainment in the former class of spaces as well as in their canonical preduals
We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct $\ell_1$-sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over balls and spheres of the same finite dimensions are isomorphic, that the Lipschitz-free space over $\mathbb{Z}^d$ is isomorphic to its $\ell_1$-sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to $\ell_1$. Moreover, following new ideas from [E. Bruè, S. Di Marino and F. Stra, Linear Lipschitz and $C^1$ extensi…
Let ϕ be an analytic self-map of D and be an analytic operator-valued function on D, where D is the unit disk. We provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators W_{ψ,ϕ} on Lip_A(D;X;α) and lip_A(D;X;α), the spaces of analytic X-valued Lipschitz functions f, where X is a complex Banach space and α is in (0, 1].
We prove that the Lipschitz-free space over a Banach space $X$ of density $κ$, denoted by $\mathcal{F}(X)$, is linearly isomorphic to its $\ell_1$-sum $\left(\bigoplus_κ\mathcal{F}(X)\right)_{\ell_1}$. This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at $0$ over a $\mathcal{L}_p$-space. More precisely, we establish that, for every $1\leq p\leq \infty$, if $X$ is a $\mathcal{L}_p$-space of density $κ$, then $\mathrm{Lip}_0(X)$ is e…
Under the right conditions on a compact metric space $X$ and on a Banach space $E$, we give a description of the $2$-local (standard) isometries on the Banach space $\hbox{Lip}(X,E)$ of vector-valued Lipschitz functions from $X$ to $E$ in terms of a generalized composition operator, and we study when every $2$-local (standard) isometry on $\hbox{Lip}(X,E)$ is both linear and surjective.
Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this…
We study a variant of the Whitney extension problem for the space $C^{k,ω}(R^n)$. We identify this space with a space of Lipschitz mappings from $R^n$ into the space $P_k \times R^n$ of polynomial fields on $R^n$ equipped with a certain metric. This identification allows us to reformulate the Whitney problem for $C^{k,ω}(R^n)$ as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of $P_k \times R^n$. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, …
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.
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.
For complete metric spaces $X$ and $Y$, a description of linear biseparating maps between spaces of vector-valued Lipschitz functions defined on $X$ and $Y$ is provided. In particular it is proved that $X$ and $Y$ are bi-Lipschitz homeomorphic, and the automatic continuity of such maps is derived in some cases. Besides, these results are used to characterize the separating bijections between scalar-valued Lipschitz function spaces when $Y$ is compact.
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.
We construct Markov semi-groups $\mathcal{T}$ and associated BMO-spaces on a finite von Neumann algebra $(\mathcal{M}, τ)$ and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that for any $A \in \mathcal{M}$ self-adjoint and $f: \mathbb{R} \rightarrow \mathbb{R}$ Lipschitz there is a Markov semi-group $\mathcal{T}$ such that for $x \in \mathcal{M}$, \[ \Vert [f(A), x] \Vert_{{\rm BMO}(\mathcal{M}, \mathcal{T})} \leq c_{abs} \Vert f' \Vert_\infty \Vert [A, x] \Vert_\infty. \] We obtain an analogue of this result for more general v…
We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$-algebras. As a consequence we confirm the statement of \cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.
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.
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.
By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe \textit{rigorously} the representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} Φ\circ | ΔX_s |$, where $X : Ω\times \R_+ \longrightarrow \R$ is a stochastic process with regulated trajectories, $t \in \R_+$ and $Φ: \R_+ \longrightarrow \R_+$ is a strictly increasing function which maps 0 to 0 (cf. P…
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 …
Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this…
Let $\frak{F}(X, A)$ be one of the Banach algebras $\hbox{Lip}(X, A)$ or $\hbox{lip}(X, A)$. In this paper, we show that $\frak{F}(X, A)$ is amenable if and only if $X$ is uniformly discrete and $A$ is amenable. We also prove that the result holds for $\hbox{lip}^\circ(X, A)$ instead of $\frak{F}(X, A)$. In the case where $A^*$ is separable, we establish that $\frak{F}(X, A)^{**}$ is amenable if and only if $X$ is uniformly discrete and $A^{**}$ is amenable, however, amenability of $\hbox{lip}^\circ(X, A)^{**}$ is equivalent to amenability of $A^{**}$ and finiteness of $X$. We prove that if $\…
Let $X, Y$ be complete metric spaces and $E, F$ be Banach spaces. A bijective linear operator from a space of $E$-valued functions on $X$ to a space of $F$-valued functions on $Y$ is said to be biseparating if $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are characterized as weighted composition operators, i.e., of the form $Tf(y) = S_y(f(h^{-1}(…
Let (X,d) be a metric space and $ α> 0 $. In this paper, we study extensions of some complex-valued Lipschitz functions, from some special subset $ X_0 $ to X. These extensions are with no-increasing Lipschitz number or the smallest Lipschitz number. Moreover, we show that under some conditions, Tietze extension theorem can be generalized for Lipschitz functions and call it Tietze-Lipschitz extension. Furthermore, we generalize Urysohn-lemma for Lipschitz functions. In fact we present a necessary and sufficient condition for that Lipschitz functions separate subsets of X.
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.
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…
Let $X$ and $Y$ be Banach or normed linear spaces and $F\subset X$ a closed set. We apply our recent extension theorem for vector-valued Baire one functions arXiv:1512.03717 to obtain an extension theorem for vector-valued functions $f\colon F\to Y$ with pre-assigned derivatives, with preservation of differentiability (at every point where the pre-assigned derivative is actually a derivative), preservation of continuity, preservation of (pointwise) Lipschitz property and (for finite dimensional domain $X$) preservation of strict differentiability and global (eventually local) Lipschitz continu…
Let $\mathrm{Lip}_0(X)$ be the space of all Lipschitz scalar-valued functions on a pointed metric space $X$. We characterize the approximation property for $\mathrm{Lip}_0(X)$ with the bounded weak* topology using as tools the tensor product, the $ε$-product and the linearization of Lipschitz mappings.
We investigate sufficient conditions for real-valued functions on product spaces to be bounded from above by sums or products of functions which depend only on points in the respective factors.
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…
Non-Lipschitz Inertial Contraction-Type Method for Monotone Variational Inclusion problems
VIT Bhopal University（维特博帕尔大学）；Aligarh Muslim University（阿利加尔穆斯林大学）；China Medical University Hospital, China Medical University（中国医药大学附设医院，中国医药大学）；Academy of Romanian Scientists（罗马尼亚科学家学院）；Tiangong University（天津工业大学）
Let $({\mathcal M},ρ)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from ${\mathcal M}$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of $F$ with the sharp value of the finiteness number.
We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of natural $C(Y)$-valuezations that take values in unital commutative $C^*$-algebras. A precise proof of a theorem of Jarosz \cite{ja} is exhibited.
In this note, we study some concentration properties for Lipschitz maps defined on Hamming graphs, as well as their stability under sums of Banach spaces. As an application, we extend a result of Causey on the coarse Lipschitz structure of quasi-reflexive spaces satisfying upper $\ell_p$ tree estimates to the setting of $\ell_p$-sums of such spaces. Our result provides us with a tool for constructing the first examples of Banach spaces that are not quasi-reflexive but nevertheless admit some concentration inequality. We also give a sufficient condition for a space to be asymptotic-$c_0$ in ter…
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.
Let $(M,ρ)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from $M$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of $F$ with the sharp value of the finiteness constant.
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 consider a certain type of geometric properties of Banach spaces, which includes for instance octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example finite $\ell^p$-sums), then it is also stable under the formation of corresponding Köthe-Bochner spaces (for example $L^p$-Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some …
In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact, limited and completely continuous linear operators. Moreover, our results yield a characterization of Gelfand-Phillips spaces and recover some known result of Schur spaces and reflexive spaces concerning the differentiability of real-valued Lipschitz…
We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n-1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.
The convex cone $SC_{\mathrm{SLip}}^1(\mathcal{X})$ of real-valued smooth semi-Lipschitz functions on a Finsler manifold $\mathcal{X}$ is an order-algebraic structure that captures both the differentiable and the quasi-metric feature of $\mathcal{X}$. In this work we show that the subset of smooth semi-Lipschitz functions of constant strictly less than $1$, denoted $SC_{1^{-}}^1(\mathcal{X})$, can be used to classify Finsler manifolds and to characterize almost isometries between them, in the lines of the classical Banach-Stone and Mykers-Nakai theorems.
It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.
On Weak Compactness and Uniform Regularity in Lipschitz free Spaces
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València（瓦伦西亚理工大学纯数学与应用数学大学研究所）；Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR8050（巴黎东克雷泰伊大学、古斯塔夫·埃菲尔大学、法国国家科学研究中心LAMA实验室）；Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623)（玛丽和路易斯·巴斯德大学、法国国家科学研究中心LmB实验室）；University of Tartu, Institute of Mathematics and Statistics（塔尔图大学数学与统计学院）；University of Innsbruck, Department of Mathematics（因斯布鲁克大学数学系）
We study a new bi-Lipschitz invariant λ(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor controlled by λ(M). We prove that λ(M) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain classes of Riemannian manifolds of bounded geometry and finite direct sums of arbitrary combinations of these objects. On the other hand we construct an ex…
We present a way to turn an arbitrary (unbounded) metric space $\mathcal{M}$ into a bounded metric space $\mathcal{B}$ in such a way that the corresponding Lipschitz-free spaces $\mathcal{F}(\mathcal{M})$ and $\mathcal{F}(\mathcal{B})$ are isomorphic. The construction we provide is functorial in a weak sense and has the advantage of being explicit. Apart from its intrinsic theoretical interest, it has many applications in that it allows to transfer many arguments valid for Lipschitz-free spaces over bounded spaces to Lipschitz-free spaces over unbounded spaces. Furthermore, we show that with a…
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
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…
Let S_k be the k-th partial sum of Banach space valued independent identically distributed random variables. In this paper, we compare the tail distribution of ||S_k|| with that of ||S_j||, and deduce some tail distribution maximal inequalities. Theorem: There is universal constant c such that for j < k Pr(||S_j|| > t) <= c Pr(||S_k|| > t/c).
We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly.
