For a metric space $X$, we study the space $D^{\infty}(X)$ of bounded functions on $X$ whose infinitesimal Lipschitz constant is uniformly bounded. $D^{\infty}(X)$ is compared with the space $\LIP^{\infty}(X)$ of bounded Lipschitz functions on $X$, in terms of different properties regarding the geometry of $X$. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare $D^{\infty}(X)$ with the Newtonian-Sobolev space $N^{1, \infty}(X)$. In particular, if $X$ supports a doubling measure and satisfies a local Poincar{é} inequality, we obtain that $D^{\infty}(X)=N^{1, \infty}(X)$.
Given any continuous, lower bounded and $κ$-convex function $V$ on a metric measure space $(X,d,m)$ which is infinitesimally Hilbertian and satisfies some synthetic lower bound for the Ricci curvature in the sense of Lott-Sturm-Villani, we prove existence and uniqueness for the (downward) gradient flow for $V$. Moreover, we prove Lipschitz continuity of the flow w.r.t. the starting point $d(x_t,x'_t)\le e^{-κ\, t} d(x_0,x_0').$
We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $μ$ on such a space, we introduce $μ$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $μ$. We show that if $μ$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $μ$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinite…
We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian $CD^*(0,N)$-spaces.
The aim of this paper is to show the existence of a canonical distance $\mathsf d'$ defined on a locally Minkowski metric measure space $(\mathsf X,\mathsf d,\mathfrak m)$ such that: i) $\mathsf d'$ is equivalent to $\mathsf d$, ii) $(\mathsf X, \mathsf d', \mathfrak m)$ is infinitesimally Hilbertian. This new regularity assumption on $(\mathsf X, \mathsf d,\mathfrak m)$ essentially forces the structure to be locally similar to a Minkowski space and defines a class of metric measure structures which includes all the Finsler manifolds, and it is actually strictly larger. The required distance $…
We provide examples of infinitesimally Hilbertian, rectifiable, Ahlfors regular metric measure spaces having pmGH-tangents that are not infinitesimally Hilbertian.
Every integral current in a locally compact metric space $X$ can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into $X$ can be extended slightly.
In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones.
We characterize uniformly perfect, complete, doubling metric spaces which embed bi- Lipschitzly into Euclidean space. Our result applies in particular to spaces of Grushin type equipped with Carnot-Carathéodory distance. Hence we obtain the first example of a sub-Riemannian mani- fold admitting such a bi-Lipschitz embedding. Our techniques involve a passage from local to global information, building on work of Christ and McShane. A new feature of our proof is the verification of the co-Lipschitz condition. This verification splits into a large scale case and a local case. These cases are disti…
M. Gromov introduced the Lipschitz order relation on the set of metric measure spaces and developed a rich theory. In particular, he claimed that an isoperimetric inequality on a non-discrete space is represented by using the Lipschitz order. We relax the definition of the Lipschitz order allowing an additive error to relate with an isoperimetric inequality on a discrete space. As an application, we obtain an isoperimetric inequality on the non-discrete $n$-dimensional $l^1$-cube by taking the limits of an isoperimetric inequality of the discrete $l^1$-cubes.
Consider a mapping $f\colon X\to Y$ between two metric measure spaces. We study generalized versions of the local Lipschitz number $\mathrm{Lip} f$, as well as of the distortion number $H_f$ that is used to define quasiconformal mappings. Using these, we give sufficient conditions for $f$ being a BV mapping $f\in BV_{\mathrm{loc}}(X;Y)$ or a Newton-Sobolev mapping $f\in N_{\mathrm{loc}}^{1,p}(X;Y)$, with $1\le p<\infty$.
Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the Lipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map from a subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ can be extended to a $CL$-Lipschitz mapping on $\mathbb{R}^m$. In this paper, we construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimension $m$. We prove that any Lipschitz mapping from a compact subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ may be extended to a Sobolev mapping on any b…
We show that rough isometries between metric spaces X, Y can be lifted to the spaces of real valued 1-Lipschitz functions over X and Y with supremum metric and apply this to their scaling limits. For the inverse, we show how rough isometries between X and Y can be reconstructed from structurally enriched rough isometries between their Lipschitz function spaces.
Let $X$ be a compact metric space and $\mathcal M_X$ be the set of isometry classes of compact metric spaces $Y$ such that the Lipschitz distance $d_L(X,Y)$ is finite. We show that $(\mathcal M_X, d_L)$ is not separable when $X$ is a closed interval, or an infinite union of shrinking closed intervals.
In a 2013 paper, Cheeger and Kleiner introduced a new type of dimension for metric spaces, the "Lipschitz dimension". We study the dimension-theoretic properties of Lipschitz dimension, including its behavior under Gromov-Hausdorff convergence, its (non-)invariance under various classes of mappings, and its relationship to the Nagata dimension and Cheeger's "analytic dimension". We compute the Lipschitz dimension of various natural spaces, including Carnot groups, snowflakes of Euclidean spaces, metric trees, and Sierpinski carpets. As corollaries, we obtain a short proof of a quasi-isometric …
We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.
For a metric measure space, we treat the set of distributions of 1-Lipschitz functions, which is called the 1-measurement. On the 1-measurement, we have a partial order relation by the Lipschitz order introduced by Gromov. The aim of this paper is to study the maximum and maximal elements of the 1-measurement with respect to the Lipschitz order. We present a necessary condition of a metric measure space for the existence of the maximum of the 1-measurement. We also consider a metric measure space that has the maximum of its 1-measurement.
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 $L_1$.
We give study the Lipschitz continuity of Möbius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.
Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous of a result proved for real valued Lipschitz maps defined on R2 by Alberti, Bianchini and Crippa in [1].
The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT(0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically s…
We prove that the Lipschitz free space over a certain type of discrete metric space has the Radon-Nikodým property. We also show that the Lipschitz free space over a complete, locally compact metric space has the Schur or approximation property whenever the Lipschitz free space over each compact subset also has this property.
This note is intended to be a supplement to the bi-Lipschitz decomposition of Lipschitz maps shown in [Sch]. We show that in the case of 1-Ahlfors-regular sets, the condition of having `Big Pieces of bi-Lipschitz Images' (BPBI) is equivalent to a Carleson condition.
Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if $f$ is a Lipschitz mapping from an open set in $\mathbb{R}^n$ onto a metric space $X$, then the topological dimension of $X$ equals $n$ if and only if $X$ has positive $n$-dimensional Hausdorff measure. We also prove an area formu…
We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.
We present an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear extension operators for Lipschitz functions.
We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of BiLipschitz functions f|_{F_i} so that the k-Hausdorff content of f([0,1]^k\setminus \cup F_i) is small. We thus generalize a theorem of P. Jones (1988) from the setting of R^d to the setting of a general metric space. This positively answers problem 11.13 in ``Fractured Fractals and Broken Dreams" by G. David and S. Semmes, or equivalently, question 9 from ``Thirty-three yes or no questions about map…
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.
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 $μ$.
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…
Given a metric space with a Borel probability measure, for each integer $N$ we obtain a probability distribution on $N\times N$ distance matrices by considering the distances between pairs of points in a sample consisting of $N$ points chosen indepenedently from the metric space with respect to the given measure. We show that this gives an asymptotically bi-Lipschitz relation between metric measure spaces and the corresponding distance matrices. This is an effective version of a result of Vershik that metric measure spaces are determined by associated distributions on infinite random matrices.
This paper deals with the study of parameter dependence of extensions of Lipschitz mappings from the point of view of continuity. We show that if assuming appropriate curvature bounds for the spaces, the multivalued extension operators that assign to every nonexpansive (resp. Lipschitz) mapping all its nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz constant) are lower semi-continuous and admit continuous selections. Moreover, we prove that Lipschitz mappings can be extended continuously even when imposing the condition that the image of the extension belongs to the…
There is a standard "word length" metric canonically associated to any set of generators for a group. In particular, for any integers a and b greater than 1, the additive group of integers has generating sets {a^i}_{i=0}^{\infty} and {b^j}_{j=0}^{\infty} with associated metrics d_A and d_B, respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers m and n such that a^m = b^n.
The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions…
We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with this metric, are not isometric, again with some possible exceptions of low genus.
In this note we show how to adjust some proofs of Koskela et. al 2003 and Jiang 2011 in order to show that in certain spaces $(X,d,μ)$, like $RCD(K,N)$-spaces, every Sobolev function with local $L^{p}$-Laplacian and $p>\dimμ$ is locally Lipschitz continuous.
We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite packing dimension can be embedded in some Euclidean space via a Lipschitz map.
We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to \bl\ equivalence. This is a metric space analog of a result due to Rohde. Our construction also works for all bounded turning metric circles; these need not be doubling. As a byproduct, we show that a metric quasicircle with Assouad dimension strictly less than two is bi-Lipschitz equivalent to a planar quasicircle.
We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which the mapping "behaves like a projection mapping" along with a "garbage set" that is arbitrarily small in an appropriate sense. Moreover, our control is quantitative, i.e., independent of both the particular mapping and the metric space it maps into. This improves a theorem of Azzam-Schul from the paper "Hard Sard", and answers a question left open in that paper. The proof uses ideas of quantitative…
We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak--Wenger, which satisfies a related maximality condition.
We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. We define min-max values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the min-max values of the limit space are larger than or equal to the upper limit of the min-max values of the currents …
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 prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.
In 1994, Jan Pelant proved that a metric property related to the notion of paracompactness called the uniform Stone property characterizes a metric space's uniform embeddability into $c_0(κ)$ for some cardinality $κ$. In this paper it is shown that coarse Lipschitz embeddability of a metric space into $c_0(κ)$ can be characterized in a similar manner. It is also shown that coarse, uniform, and bi-Lipschitz embeddability into $c_0(κ)$ are equivalent notions for normed linear spaces.
We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free space $\mathcal{F}(M)$ over a compact metric space $M$ is a dual space if and only if $M$ is purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodým property (RNP) and deduce that, for any complete metric space $M$, pure …
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 introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spaces without collapsing.
