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, we improve estimates for finiteness numbers in finiteness theorems for $C^{k,ω}(R^n)$ due to C. Fefferman.
We study a variant of the Whitney extension problem for the space $C^kΛ^m_ω(R^n)$ of functions whose partial derivatives of order $k$ satisfy the generalized Zygmund condition. We identify $C^kΛ^m_ω(R^n)$ with a space of Lipschitz mappings from a metric space $(R^{n+1}_+,ρ_ω)$ supplied with a hyperbolic metric $ρ_ω$ into a metric space $({\cal P}_{k+m-1}\times R^{n+1}_+, d_ω)$ of polynomial fields on $R^{n+1}_+$ equipped with a hyperbolic-type metric $d_ω$. This identification allows us to reformulate the Whitney problem for $C^kΛ^m_ω(R^n)$ as a Lipschitz selection problem for set-valued mappi…
In this paper, we give analytic methods for finding m (and m+ω) times continuously differentiable solutions of a finite system of linear equations. Along the way, we also solve a generalized Whitney problem for vector-valued functions and jets.
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 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
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].
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 construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and $C^1$ functions. This way we prove more directly a result by Lee and Naor and we generalize the $C^1$ extension theorem by Whitney to Banach spaces.
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.
Let $L^m_p(R^n)$, $p\in [1,\infty]$, be the homogeneous Sobolev space, and let $E\subset R^n$ be a closed set. For each $p>n$ and each non-negative integer $m$ we give an intrinsic characterization of the restrictions to $E$ of $m$-jets generated by functions $F\in L^{m+1}_p(R^n)$. Our trace criterion is expressed in terms of variations of corresponding Taylor remainders of $m$-jets evaluated on a certain family of "well separated" two point subsets of $E$. For $p=\infty$ this result coincides with the classical Whitney-Glaeser extension theorem for $m$-jets. Our approach is based on a represe…
Order-Preserving Extensions of Hadamard Space-Valued Lipschitz Maps
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Department of Economics and Courant Institute of Mathematical Sciences, New York University（纽约大学经济学系与库朗数学科学研究所）
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.
Let $d$ be a metric on $R^n$ and let $C^{m,(d)}(R^n)$ be the space of $C^m$-function on $R^n$ whose partial derivatives of order $m$ belong to the space $Lip(R^n;d)$. We show that the homogeneous Sobolev space $L^{m+1}_p(R^n),p>n,$ can be represented as a union of $C^{m,(d)}(R^n)$-spaces where $d$ belongs to a family of metrics on $R^n$ with certain "nice" properties. This enables us in several important cases to give intrinsic characterizations of the restrictions of Sobolev spaces to arbitrary closed subsets of $R^n$. In particular, we generalize the classical Whitney extension theorem for t…
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.
This paper addresses Lipschitzian stability issues that play an important role in both theoretical and numerical aspects of variational analysis, optimization, and their applications. We particularly concentrate on the so-called relaxed one-sided Lipschitz property of set-valued mappings with negative Lipschitz constants. This property has been much less investigated than more conventional Lipschitzian behavior while being well recognized in a variety of applications. Recent work has revealed that set-valued mappings satisfying the relaxed one-sided Lipschitz condition with negative Lipschitz …
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.
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.
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 $\mathrm{Lip}_0(X)$ 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 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 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 $X$ 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.
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.
We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allo…
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.
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…
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.
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…
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…
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 $\ell^{\infty}$.
Let $X$ and $Y$ be compact Hausdorff spaces, $E$ and $F$ be real or complex normed spaces and $A(X,E)$ be a subspace of $C(X,E)$. For a function $f\in C(X,E)$, let $\coz(f)$ be the cozero set of $f$. A pair of additive maps $S,T: A(X,E) \lo C(Y,F)$ is said to be jointly separating if $\coz(Tf)\cap \coz(Sg)=\emptyset$ whenever $\coz(f)\cap \coz(g)= \emptyset$. In this paper, first we give a partial description of additive jointly separating maps between certain spaces of vector-valued continuous functions (including spaces of vector-valued Lipschitz functions, absolutely continuous functions an…
We provide sufficient conditions for a set $E\subset\mathbb{R}^n$ to be a non-universal differentiability set, i.e. to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of $\mathbb{R}^n$ given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set $E$ in $\mathbb{R}^n$ there is a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^n$ not differentiable at any point …
We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say $x$ and $y$, which define it satisfy that the metric segment $[x, y]$ only contains $x$ and $y$. The most notable among them is the case when the free space admits an isometric predual with some additional p…
