Kevin Wildrick, Thomas Zürcher
We give a sharp condition on the lower local Lipschitz constant of a mapping from a metric space supporting a Poincaré inequality to a Banach space with the Radon-Nikodym property that guarantees differentiability at almost every point. We apply these results to obtain a non-embedding theorem for a corresponding class of mappings.
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We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Banach space. In contrast, we also show that if an Ahlfors regular Lipschitz differentiability space has charts of non-maximal dimension, then these charts are strongly unrectifiable in the sense of Ambrosio-Kirchh…
We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that $l^2$-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Nikodym property which fail to be differentiable on sets of positive measure.
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space…
Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type, where the Lebesgue measure is replaced by an arbitrary measure $μ$. In particular we show that the differentiability properties of Lipschitz functions at $μ$-almost every point are related to the decompositions of $μ$ in terms of rectifiable one-dimensional measures. As a consequence we obtain a differentiability result for Lipschitz functions with respect …
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 prove metric differentiation for differentiability spaces in the sense of Cheeger. As corollaries we give a new proof that the minimal generalized upper gradient coincides with the pointwise Lipschitz constant for Lipschitz functions on PI spaces, a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith is equal to $1$, and new nonembeddability results.
We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of Lipschitz curves in a way analogous to the differentiability theory of Euclidean spaces. This approach to differentiability in this generality appears here for the first time and by examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.
We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$-weak gradient on iterated blow-ups of differentiability spaces.
In this paper we prove that the typical Lipschitz function has no directional derivative at any point of a Borel set $E$ if and only if $E$ is contained in a countable union of closed purely unrectifiable sets.
Embeddability and rectifiability of Lipschitz differentiability spaces
本文通过可分解束和爆破分析，证明了双Lipschitz嵌入RNP空间的Lipschitz可微空间可数可修正，并给出新的刻画及解决Le Donne问题。
Shortcut Laakso spaces, pure PI unrectifiability and differentiability of Lipschitz functions
该研究构造了纯PI不可 rectifiable 的Lipschitz可微空间族，探究捷径度量空间的几何并刻画其PI可 rectifiability 与Y-LDS性质，验证Laakso空间满足相关假设。
Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas
本文自包含地阐述了欧几里得空间子集到任意度量空间的Lipschitz映射的度量可微性，并利用新的度量隐函数定理，直接且几何地证明了相应的面积公式和余面积公式。
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 use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a $σ$-porous set. The second result states that irregular points of a Lipschitz function form a $σ$-porous set. We use these observations to give a new proof of Pansu's theorem for Lipschitz maps from a general Carnot group to a Euclidean space.
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.
Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings
本文提出广义局部Lipschitz数，用于统一刻画Sobolev函数与有界变差函数，并建立精细可微性、拟共形映射、Alberti秩一定理等多个领域之间的新联系。
A new characterization of Sobolev spaces on Lipschitz differentiability spaces
本文在不假设双重条件或Poincaré不等式的情况下，通过证明精确的Brezis-Van Schaftingen-Yung型渐进行式，给出了Sobolev范数通过非局部泛函渐进行为的新的刻画，并构造了反例以证明假设的必要性。
We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from a recent article by Bate--Kangasniemi--Orponen, this equivalence is used to show that the ${\rm Lip}-{\rm lip}$ -type condition ${\rm lip} f\le C|Df|$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to ${\rm lip} f =|Df|$. We also provide a direct proof of a result…
We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon-Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect near by points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger-Kleiner, we show that our conditions are also sufficient. We also develop another characterization of "RNP Lipschitz differenti…
Connections between metric differentiability and rectifiability
结合 Kirchheim 度量微分与 Cheeger 图表，确立了 Banach 或度量空间族的不可嵌入原理，并以此证明 Lipschitz 可微性空间可整流性及 Kirchheim 定理的逆定理。
Carnot rectifiability and Alberti representations
本文通过阿尔贝蒂表示与取值于卡诺群的李普希茨映射可微性，刻画了卡诺可整流性，并提出潘苏可微性空间（PDS）作为切格尔可微性空间的卡诺群类比。
This paper contributes to the generalization of Rademacher's differentiability result for Lipschitz functions when the domain is infinite dimensional and has nonabelian group structure. We introduce the notion of metric scalable groups which are our infinite-dimensional analogues of Carnot groups. The groups in which we will mostly be interested are the ones that admit a dense increasing sequence of (finite-dimensional) Carnot subgroups. In fact, in each of these spaces we show that every Lipschitz function has a point of Gâteaux differentiability. We provide examples and criteria for when suc…
Bochner Partial Derivatives, Cheeger-Kleiner Differentiability, and Non-Embedding
该研究在庞加莱不等式空间中定义含次黎曼Heisenberg群的Cheeger分形类，证明此类分形无法双Lipschitz嵌入满足特定Bochner偏导数条件的Banach空间，拓展了Creutz和Evseev的相关成果。
This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show that the push-forward of the measure from a Lipschitz differentiability space under a chart is absolutely continuous with respect to Lebesgue measure.
Weak derivatives and metric differentiability almost everywhere
本文针对任意度量或Banach空间中Lipschitz映射不一定弱*可微的问题，提出基于弱弱*导数的方法，为度量微分提供线性表示并使其可通过线性算子范数计算。
Universal Differentiability Sets in Laakso Space
本文在 Laakso 空间中构造一族相互奇异的加倍测度，使实值 Lipschitz 函数几乎处处可微，并由此得到测度为零的通用可微性集合，同时证明这些测度均支持 Poincaré 不等式。
We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a $σ$-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a $σ$-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces and Carnot groups.
Suppose that $(X,d,μ)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincaré inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbour…
The Stepanov theorem for Q-valued functions
本文在度量空间多值函数的一般框架下证明了Stepanov微分定理，并引入了可微性与近似可微性的通用定义，无需Lipschitz扩张条件。
Stepanov theorem for mappings between metric spaces
本文在度量空间之间提出一种基于可数图表的度量可微性概念，并证明了其在相同假设下的斯蒂潘诺夫型推广的正确性。
On metric spaces equipped with doubling measures, we prove that a differentiability theorem holds for Lipschitz functions if and only if the space supports nontrivial (metric) derivations in the sense of Weaver that satisfy an additional infinitesmal condition. In particular it extends the case of spaces supporting Poincaré inequalities, as first proven by Cheeger, as well as the case of spaces satisfying the Lip-lip condition of Keith. The proof relies on generalised "change of variable" arguments that are made possible by the linear algebraic structure of derivations. As a crucial step in th…
Martingale and analytic dimensions
本文定义了鞅维数与解析维数，证明在双侧高斯热核估计下二者一致，并引入能量像密度性质，最后指出行走维数大于2的次高斯扩散空间非利普希茨可微。
This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X-->V, and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V=L^1 where differentiability fails. We establish another kind of differentiability for certain X, including R^n and H, the Heisenberg group with its Carnot-Cartheodory metric. It follows that H does not bi-Lipschitz embed into L^1, as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counter example to the Goemans-Linial…
Recently Gigli developed a Sobolev calculus on non-smooth spaces using module theory. In this paper it is shown that his theory fits nicely into the theory of differentiability spaces initiated by Cheeger, Keith and others. A relaxation procedure for $L^p$-valued subadditive functionals is presented and a relationship between the module generated by a functional and the one generated by its relaxation is given. In the framework of differentiability spaces, which includes so called PI- and $RCD(K,N)$-spaces, the Lipschitz module is pointwise finite dimensional. A general renorming theorem toget…
If $X$ is a subset of a Banach space with $X-X$ homogeneous, then $X$ can be embedded into some $\R^n$ (with $n$ sufficiently large) using a linear map $L$ whose inverse is Lipschitz to within logarithmic corrections. More precisely, $$c\,\frac{\|x-y\|}{|\,\log\|x-y\|\,|^α}\le|Lx-Ly|\le c\|x-y\|$$ for all $x,y\in X$ with $\|x-y\|<δ$ for some $δ$ sufficiently small. A simple argument shows that one must have $α>1$ in the case of a general Banach space and $α>1/2$ in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banac…
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