On Rectifiable Measures in Carnot Groups: Existence of Density

  title={On Rectifiable Measures in Carnot Groups: Existence of Density},
  author={Gioacchino Antonelli and Andrea Merlo},
  journal={Journal of Geometric Analysis},
In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is Ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {P}}_h$$\end{document}-rectifiable, for h∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage… 

Nowhere differentiable intrinsic Lipschitz graphs

We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow‐up limits, none of which is a homogeneous subgroup. This

Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N .

Parabolic rectifiability, tangent planes and tangent measures

We define rectifiability in R ×R with a parabolic metric in terms of C graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and

Normed spaces using intrinsically Lipschitz sections and Extension Theorem for the intrinsically H\"older 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

Non-symmetric intrinsic Hopf-Lax semigroup vs. intrinsic Lagrangian

. In this paper, we analyze the ’symmetrized’ of the intrinsic Hopf-Lax semigroup introduced by the author in the context of the intrinsically Lipschitz sections in the setting of metric spaces.

On the converse of Pansu's Theorem

. We provide a suitable generalisation of Pansu’s differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are

Intrinsically H\"older sections in metric spaces

. 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,

On sets with unit Hausdorff density in homogeneous groups

It is a longstanding conjecture that given a subset E of a metric space, if E has finite Hausdorff measure in dimension α and H α (cid:120) E has unit density almost everywhere, then E is an α

On rectifiable measures in Carnot groups: Marstrand–Mattila rectifiability criterion

Identifying 1-rectifiable measures in Carnot groups

. We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem



A rectifiability result for finite-perimeter sets in Carnot groups

In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is possibly,

On uniform measures in the Heisenberg group

Characteristic points, rectifiability and perimeter measure on stratified groups

We establish an explicit formula between the perimeter measure of an open set $E$ with $C^1$ boundary and the spherical Hausdorff measure $\cS^{Q-1}$ restricted to $\der E$, when the ambient space is

Semigenerated step-3 Carnot algebras and applications to sub-Riemannian perimeter

This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical

Metric Lie groups admitting dilations

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$,

Pliability, or the whitney extension theorem for curves in carnot groups

The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given

Lipschitz graphs and currents in Heisenberg groups

  • D. Vittone
  • Mathematics
    Forum of Mathematics, Sigma
  • 2022
Abstract The main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups ${\mathbb H}^{n}$ . For

Intrinsically Lipschitz functions with normal target in Carnot groups

We provide a Rademacher theorem for intrinsically Lipschitz functions $\phi:U\subseteq \mathbb W\to \mathbb L$, where $U$ is a Borel set, $\mathbb W$ and $\mathbb L$ are complementary subgroups of a

Marstrand-Mattila rectifiability criterion for $1$-codimensional measures in Carnot Groups

This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with

Regular Hypersurfaces, Intrinsic Perimeter and Implicit Function Theorem in Carnot Groups

In the last few years, a systematic attempt to develop geometric measure theory in metric spaces has become the object of many studies. Such a program, already suggested in Federer’s book [17], has