Intrinsically Lipschitz graphs on semidirect products of groups

A note about Intrinsically Lipschitz constants

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.

本征Lipschitz截面及其在度量群中的应用

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。

Normed spaces using intrinsically Lipschitz sections and Extension Theorem for the intrinsically Hölder sections

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 $Y.$

Intrinsic Cheeger energy for the intrinsically Lipschitz constants

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…

Intrinsically Lipschitz functions with normal target in Carnot groups

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.

Intrinsic Lipschitz sections of no-linear quotient maps

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…