Daniela Di Donato
Recently, Le Donne and the author introduce a notion of intrinsically Lipschitz graphs in metric spaces. The idea of this paper is to investigate about the properties of the intrinsically Lipschitz constants. More precisely, we give the Leibniz formula and the product formula for the intrinsic slope.
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Recently, in the metric spaces, Le Donne and the author introduced the so-called intrinsically Lipschitz sections. The main aim of this note is to adapt Cheeger theory for the classical Lipschitz constants in our new context. More precisely, we define the intrinsic Cheeger energy from $L^2(Y,\R^s)$ to $[0,+\infty],$ where $(Y,d_Y,\mm)$ is a metric measure space and we characterize it in terms of a suitable notion of relaxed slope. In order to get this result, in more general context, we establish some properties of the intrinsically Lipschitz constants like the Leibniz formula, the product for…
In the metric spaces, we give some equivalent condition of intrinsically Lipschitz maps introduce by Franchi, Serapioni and Serra Cassano in subRiemannian Carnot groups. Unlike what happens in the Carnot groups, in our context intrinsic dilation do not exist but we can prove the same results using the Lipschitz property of the projection maps.
We provide a Rademacher theorem for intrinsically Lipschitz functions $ϕ:U\subseteq \mathbb W\to \mathbb L$, where $U$ is a Borel set, $\mathbb W$ and $\mathbb L$ are complementary subgroups of a Carnot group, where we require that $\mathbb L$ is a normal subgroup. Our hypotheses are satisfied for example when $\mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.
The purpose of this article is twofold: first of all, we want to define two norms using the space of intrinsically Lipschitz sections. On the other hand, we want to generalize an Extension Theorem proved by the author in the context of the intrinsically Hölder sections with target a topological space $Y.$ Here our target will be $Y\times \R^s$ with $s \geq 1$ instead of $Y.$
Intrinsically Lipschitz sections and applications to metric groups
该研究在度量空间中提出本征Lipschitz截面的广义概念，证明了相关紧性、正则性及延拓定理，将Carnot群中的已有经典结果纳入为推论。
Le Donne and the author introduced the so-called intrinsically Lipschitz sections of a fixed quotient map $π$ in the context of metric spaces. Moreover, the author introduced the concept of intrinsic Cheeger energy when the quotient map is also linear. In this note we investigate about the non linearity of $π$. In particular, we find a Leibniz formula for the intrinsic slope when $π$ satisfies a weaker condition. After that, we focus our attention on Carnot groups and using the properties of intrinsic dilations we show that the dilation of a Lipschitz section is so too. Finally, in Carnot grou…
We introduce a definition of intrinsically quasi-symmetric sections in metric spaces and we prove the Ahlfors-David regularity for this class of sections. We follow a recent result by Le Donne and the author where we generalize the notion of intrinsically Lipschitz graphs in the sense of Franchi, Serapioni and Serra Cassano. We do this by focusing our attention on the graph property instead of the map one.
We introduce a notion of intrinsically Hölder graphs in metric spaces. Following a recent paper of Le Donne and the author, we prove some relevant results as the Ascoli-Arzelà compactness Theorem, Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over $\R$ or $\C,$ a convex set and an equivalence relation for intrinsically Hölder graphs. These last three properties are new also in the Lipschitz case. Throughout the paper, we use basic mathematical too…
The coarea formula for projections of Lipschitz mappings on Carnot groups
本文证明了Carnot群上内在Lipschitz映射沿水平投影的余面积不等式，并在有限余畸变映射及Heisenberg群情形下取得等式。
We address the question of the bi-Lipschitz local triviality of a complex polynomial function over a complex value. Our main result state that a non constant complex polynomial admits a locally bi-Lipschitz trivial value if and only if it is a polynomial in a single complex variable.
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].
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.
This note is a contribution to large scale geometry. More precisely, we introduce the intrinsically quasi-isometric sections in metric spaces and we investigate their properties: the Ahlfors-David regularity in large scale; following Cheeger theory, it is possible to define suitable sets in order to obtain convexity and being a vector space over $\mathbb{R}$ or $\mathbb{C}$ for these sections; yet, following Cheeger's idea, we give an equivalence relation for this class of sections. Throughout the paper, we use basic mathematical tools.
Inspired by a recent work of Wang-Zhao, in this note we prove that for a fixed $n$-dimensional closed Riemannian manifold $(M^n, g)$, if an $\mathrm{RCD}(K, n)$ space $(X, \mathsf{d}, \mathfrak{m})$ is Gromov-Hausdorff close to $M^n$, then there exists a regular homeomorphism $F$ from $X$ to $M^n$ such that $F$ is Lipschitz continuous and that $F^{-1}$ is Hölder continuous, where the Lipschitz constant of $F$, the Hölder exponent and the Hölder constant of $F^{-1}$ can be chosen arbitrary close to $1$. This is sharp in the sense that in general such a map cannot be improved to being bi-Lipschi…
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.
Sampaio recently showed that bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones. The purpose of this note is to show that Sampaio's method works as well for (SSP) sets, that is, the above result is characteristic for (SSP) sets, a much wider class.
Lipschitz distance between clouds
本文证明了单点度量空间和实数线到有限Gromov-Hausdorff距离的度量空间类的Lipschitz距离为正，揭示了这些类之间的分离性。
We introduce the sketch of construction of a Lipschitz function $f: \mathcal{C} \to H$, here the set $\mathcal{C}$ consists of all compact convex subsets of the Hadamard space $H$. Note that $f(\{x\})=\{x\}$ for any point $x \in H$.
In this note, we introduce a natural notion of intrinsic Hopf-Lax semigroup in the context of the so-called intrinsically Lipschitz sections. The main aims are to prove the link between the intrinsic Hopf-Lax semigroup and the intrinsic slope and to show that the intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.
A note on absolutely minimal extensions in finite metric spaces
该研究探讨了有限度量空间中绝对最小Lipschitz扩展的存在性，指出在五点空间中此类扩展可能无法成立。
A note on Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces
本文研究了坍缩RCD(K,N)-空间上的伯努利型自由边界问题，证明了极小解的存在性及其局部Lipschitz连续性，并展示了自由边界具有局部有限perimeter。
This short note contains an elementary observation in response to the recent posting arXiv:1707.06593v1, which studies the Lipschitz extension modulus to $n$ additional points. We bound this modulus in terms of the well-studied Lipschitz extension modulus from $n$ points to the entire ambient metric space, thus making it possible to quote the available literature to improve some of the bounds obtained in arXiv:1707.06593v1.
We give a proof, based on Lipschitz quotient mappings, for the fact that limits of BLD-mappings between manifolds of bounded geometry are BLD. Furthermore we show that such mappings share some properties of covering maps and especially have no asymptotic values.
On the constant of Lipschitz approximability
本研究找到λ>1并显式构造出一个可分Banach空间X，证明不存在从X到其闭线性张成空间为X的紧凸子集的λ-Lipschitz收缩映射，是首个具备该性质的例子，与2014年Godefroy和Ozawa提出的公开问题密切相关。
Smooth approximations preserving asymptotic Lipschitz bounds
本文证明了 Banach 空间上的实值 Lipschitz 函数可在 $σ$-紧集上被渐近 Lipschitz 常数受控的光滑柱函数逐点逼近，并由此推出光滑柱函数在加权 Banach 空间上的度量 Sobolev 与 BV 空间中按能量稠密。
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.
These notes have the intent to introduce the study of the nonlinear aspects of operator space theory. We investigate some results on the nonlinear theory of Banach spaces which remain valid in the noncommutative case. In particular, we show that Ribe's theorem has a complete analog to preduals of von Neumann algebras and introduce the concept of the Lipschitz free operator space of an operator space. We use those to prove results about injective von Neumann algebras, Pisier's operator space OH, and the existence of complete linear isometric embeddings between operator spaces.
The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson-type estimate.
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