Michael Doré, Olga Maleva
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.
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We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of sigma-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.
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 …
We verify the existence of a purely unrectifiable set in which the typical Lipschitz function has a large set of full differentiability points. The example arises from a construction, due to Csörnyei, Preiss and Tišer, of a universal differentiability set in which a certain Lipschitz function has only a purely unrectifiable set of differentiability points.
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 …
In this note we study the differentiability with respect to the time-parameter of semigroups consisting of Lipschitzian or smooth self-mappings of a domain in a Banach space.
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…
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.
We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space.
We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form $μ=\sum_n λ_n \frac{δ_{x_n}-δ_{y_n}}{d(x_n,y_n)}$ such that $\|μ\|=\sum_n |λ_n |$. We characterise these elements in terms of geometric conditions on the points $x_n$, $y_n$ of the underlying metric space, and determine when they are points of Gâteaux differentiability of the norm. In particular, we show that Gâteaux and Fréchet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and that their tensor products with G…
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。
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.
Extreme non-differentiability of typical Lipschitz mappings
研究显示，无论给定规范空间的子集如何，典型1-Lipschitz映射在强意义上在集合的典型点处非可导，且在任意小尺度上导数比近似任意线性算子。
Differentiability of Lipschitz curves in $p$-Banach spaces
本文刻画了拟Banach空间中每条Lipschitz曲线可微的充要条件为同构于具有Radon–Nikodým性质的Banach空间，并研究$\ell_p$（$0<p<1$）中解耦坐标构造下的几乎处处可微性。
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。
A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called $Γ$-null sets), whose definition involves both the notions of category…
We study the relationships between Gateaux, weak Hadamard and Frechet differentiability and their bornologies for Lipschitz and for convex functions. In particular, Frechet and weak Hadamard differentiabily coincide for all Lipschitz functions if and only if the space is reflexive (an earlier paper of the first two authors shows that these two notions of differentiability coincide for continuous convex functions if and only if the space does not contain a copy of $\ell_1$). We also examine when Gateaux and weak Hadamard differentiability coincide for continuous convex functions. For instance, …
We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f:X\to\R$ cone mo…
Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ a mapping. We prove that there exists a $σ$-directionally porous set $A\subset X$ such that if $x\in X \setminus A$, $f$ is Lipschitz at $x$, and $f$ is Gâteaux differentiable at $x$, then $f$ is Hadamard differentiable at $x$. If $f$ is Borel measurable (or has the Baire property) and is G\^ ateaux differentiable at all points, then $f$ is Hadamard differentiable at all points except a set which is $σ$-directionally porous set (and so is Aronszajn null, Haar null and $Γ$-null). Consequently, an everywhere G\^ ateaux diff…
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…
Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ an arbitrary mapping. Then the following implication holds at each point $x \in X$ except a $σ$-directionally porous set: If the one-sided Hadamard directional derivative $f'_{H+}(x,u)$ exists in all directions $u$ from a set $S_x \subset X$ whose linear span is dense in $X$, then $f$ is Hadamard differentiable at $x$. This theorem improves and generalizes a recent result of A.D. Ioffe, in which the linear span of $S_x$ equals $X$ and $Y = \R$. An analogous theorem, in which $f$ is pointwise Lipschitz, and which deals wit…
We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has zero length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: In any such coverable set a typical Lipschitz function is everywhere severely non-differentiable.
Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fr\' echet differentiable but is not Fr\' echet differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first category set (resp. a $σ$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is Fr\…
We prove that universal differentiability sets in Euclidean spaces possess distinctive structural properties. Namely, we show that any universal differentiability set contains a `kernel' in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets. The sharpness of this result, with respect to existing decomposibility results of the opposite nature, is discussed.
We show that every finite-dimensional Euclidean space contains compact universal differentiability sets of upper Minkowski dimension one. In other words, there are compact sets $S$ of upper Minkowski dimension one such that every Lipschitz function defined on the whole space is differentiable inside $S$. Such sets are constructed explicitly.
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…
We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fréchet differentiable somewhere in S.
One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space $C[0,1]$. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull $B\supseteq A$ and a continuous map $f\colon \{0,1\}^\mathbb{N}\to C[0,1]$ such that $f^{-1}[B+h]$ is Lebesgue's null for all $h\in C[0,1]$. We prove that $\mathcal{SD}$ is not Haar-countable (i.e., does not satisfy the above property with "Lebesgue's null" re…
This paper deals with the extension of a classical theorem by R. Phelps on the Gâteaux differentiability of Lipschitz functions on separable Banach spaces to the non-separable case. The extension of the theorem is not possible for general non-separable Banach spaces, as shown by Larman. Therefore, we will work on some important non-separable spaces. We consider a norm of the non-separable space $\ell^{\infty}(\mathbb{R})$, showing that it complies with the aforementioned theorem thesis. We also consider other examples as $L^{\infty}(\mathbb{R})$ and $NBV[a,b]$, showing in all cases their sets …
We show that every model filiform group $\mathbb{E}_{n}$ contains a measure zero set $N$ such that every Lipschitz map $f\colon \mathbb{E}_{n}\to \mathbb{R}$ is differentiable at some point of $N$. Model filiform groups are a class of Carnot groups which can have arbitrarily high step. Essential to our work is the question of whether existence of an (almost) maximal directional derivative $Ef(x)$ in a Carnot group implies differentiability of a Lipschitz map $f$ at $x$. We show that such an implication is valid in model Filiform groups except for a one-dimensional subspace of horizontal direct…
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。
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。
We show that the Heisenberg group $\mathbb{H}^n$ contains a measure zero set $N$ such that every Lipschitz function $f\colon \mathbb{H}^n \to \mathbb{R}$ is Pansu differentiable at a point of $N$. The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of $N$ and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.
We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set' N such that every real-valued Lipschitz map on G is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.
We discuss removability problems concerning differentiability and pointwise Lipschitz conditions for functions of a real variable. We prove that, in each of the settings under consideration, a set is removable if and only if it has no perfect subsets.
We prove that functions of locally bounded deformation on $\mathbb{R}^n$ are $\mathrm{L}^{n/(n-1)}$-differentiable almost everywhere. More generally, we show that this critical $\mathrm{L}^p$-differentiability result holds for functions of locally bounded $\mathbb{A}$-variation, provided that the first order, homogeneous, linear differential operator $\mathbb{A}$ has finite dimensional null-space.
Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class of functions but still with Lipschitz flavour. As an application we obtain a Banach-Stone-like theorem. On the other hand, we give an extension of a result of Bourgain and Diestel related to limited operators and cosingularity.
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.
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。
A generalization of the Flow-box Theorem is given. The assumption of continuous differentiability of the vector field is relaxed to a local Lipschitz condition. The theorem holds in any Banach space.
This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). In contrast to classical results, our method of proof does not rely on Attouch's theorem on the characterization of Mosco convergence but …
Let $S$ be a closed subset of a Banach space $X$. Assuming that $S$ is epi-Lipschitzian at $\bar{x}$ in the boundary $ \bd S$ of $S$, we show that $S$ is strictly Hadamard differentiable at $\bar{x}$ IFF the Clarke tangent cone $T(S, \bar{x})$ to $S$ at $\bar{x}$ contains a closed hyperplane IFF the Clarke tangent cone $T(\bd S, \bar{x})$ to $\bd S$ at $\bar{x}$ is a closed hyperplane. Moreover when $X$ is of finite dimension, $Y$ is a Banach space and $g: X \mapsto Y$ is a locally Lipschitz mapping around $\bar{x}$, we show that $g$ is strictly Hadamard differentiable at $\bar{x}$ IFF $T(\mat…
From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability
本文研究Hilbert空间中由强单调Lipschitz单值映射和极大单调集值映射支配的广义方程，通过单值算子和预解式的方向可微性验证解映射的方向可微性，并应用于拟广义方程。
Chain rule for pointwise Lipschitz mappings
本文提出逐点 Lipschitz 映射的几乎处处链式法则，去除全局可微假设，并推广至无限维空间，同时加强 Rademacher-Stepanov 定理。
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 prove an implicit function theorem for C^k-maps from arbitrary topological vector spaces over valued fields to Banach spaces (for k at least 2). As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of contractions of a Banach space. Similar results are obtained for k times strictly differentiable maps, and for k times Lipschitz differentiable maps. In the real case, our results subsume an implicit function theorem for Keller C^k_c-maps from arbitrary topological vector spaces to Banach spaces.
We show that there is an operator space notion of Lipschitz embeddability between operator spaces which is strictly weaker than its linear counterpart but which is still strong enough to impose linear restrictions on operator space structures. This shows that there is a nontrivial theory of nonlinear geometry for operator spaces and it answers a question in [BCD21]. For that, we introduce the operator space version of Lipschitz-free Banach spaces and prove several properties of it. In particular, we show that separable operator spaces satisfy a sort of isometric Lipschitz-lifting property in t…
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…
A note on nonseparable Lipschitz-free spaces
本文研究 Lipschitz-free 空间中经典 Banach 空间性质与可分离性的等价性，指出部分性质下等价失效、相关 weak* 序列紧球问题在 ZFC 中不可判定，并构造缺失 Radon–Nikodým 性质的反例。
We investigate the differentiability properties of real-valued quasiconvex functions f defined on a separable Banach space X. Continuity is only assumed to hold at the points of a dense subset. If so, this subset is automatically residual. Sample results that can be quoted without involving any new concept or nomenclature are as follows: (i) If f is usc or strictly quasiconvex, then f is Hadamard differentiable at the points of a dense subset of X (ii) If f is even, then f is continuous and Gateaux differentiable at the points of a dense subset of X. In (i) or (ii), the dense subset need not b…
The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreĭn trace formula derived. The aim of the present paper is to prove the Frechet differe…
From the point of view of unification of differentiation theory, it is of interest to note that the general construction principle of Bertram, Gloeckner and Neeb leading to a C^k differentiability concept from a given C^0 one, besides subsuming the Keller--Bastiani C_c^k differentiabilities on real Hausdorff locally convex spaces, also does the same to the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl, and likewise to the "compactly generated" theory of Seip's continuous differentiabilities. In this article, we give the details of the proof f…
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…
We prove that the order $k$ differentiability classes for $k=0,1,...\infty$ in the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl can be obtained as particular cases of the general construction by Bertram, Glockner and Neeb leading to $C^k$ differentiabilities from a given $C^0$ concept.
Let n>2 and X be a Banach space of dimension strictly greater than n. We show there exists a directionally porous set P in X for which the set of C^1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable this leads to a decomposition of X into a countable union of directionally porous sets and a set which is null on residually many C^1 surfaces of dimension n. This is of interest in the study of certain classes of null sets used to investigate differentiability of Lipschitz functions on Banach spaces.
In this paper, we consider the inverse problem of recovering a diffusion and absorption coefficients in steady-state optical tomography problem from the Neumann-to-Dirichlet map. We first prove a Global uniqueness and Lipschitz stability estimate for the absorption parameter provided that the diffusion is known. Then, we prove a Lipschitz stability result for simultaneous recovery of diffusion and absorption. In both cases the parameters belong to a known finite subspace with a priori known bounds. The proofs relies on a monotonicity result combined with the techniques of localized potentials.…
The wark is a contribution to the functional calculus constructed by of the author and A. A. Atvinovskii of closed operators on Banach spaces. The calculus is based on Markov and related functions as symbols. Estimates of bounded perturbations of operator functions with respect to general operator ideal norms, operator Lipschitzness, moment inequality, Freshet operator differentiability, and Livshitz-Krein trace formulae have been considered.
We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C^{k}_{\mathbb{G}}$ maps. We also prove that existence of an approximate (k-1)-Taylor polynomial almost everywhere is equivalent to admitting a Lusin approximation by maps in a suitable Lipschitz function space.
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