Hostname: page-component-cd4964975-598jt Total loading time: 0 Render date: 2023-04-02T13:09:41.267Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Lipschitz graphs and currents in Heisenberg groups

Published online by Cambridge University Press:  02 February 2022

Davide Vittone*
Università di Padova, Dipartimento di Matematica ‘T. Levi-Civita’, via Trieste 63, 35121 Padova, Italy; E-mail:


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 the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between ${\mathbb H}$ -rectifiability and ‘Lipschitz’ ${\mathbb H}$ -rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.

Differential Geometry and Geometric Analysis
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

The celebrated Rademacher’s theorem [Reference Rademacher82] states that a Lipschitz continuous function $f:{\mathbb R}^{h}\to {\mathbb R}^{k}$ is differentiable almost everywhere in ${\mathbb R}^{h}$ ; in particular, the graph of f in ${\mathbb R}^{h+k}$ has an h-dimensional tangent plane at almost all of its points. One of the consequences of Rademacher’s theorem is the following Lusin-type result, which stems from Whitney’s extension theorem [Reference Whitney96]: for every $\varepsilon>0$ , there exists $g\in C^{1}({\mathbb R}^{h},{\mathbb R}^{k})$ that coincides with f out of a set of measure at most $\varepsilon $ . From the viewpoint of geometric measure theory, this means that Lipschitz-regular objects (functions, submanifolds, etc.) are essentially as nice as $C^{1}$ -smooth ones and has profound implications; for instance, in the theory of rectifiable sets and currents [Reference Federer42, Reference Mattila72, Reference Simon90].

The present article aims to develop a similar theory for submanifolds with (intrinsic) Lipschitz regularity in sub-Riemannian Heisenberg groups. Before introducing our results, we feel the need to list them at least quickly. We believe that our main result is a Rademacher-type theorem for intrinsic Lipschitz graphs, which was the main open problem since the beginning of this theory. Some applications – namely, a Lusin-type result and an area formula for intrinsic Lipschitz graphs – are provided here as well; however, we believe that further consequences are yet to come concerning, for instance, rectifiability and minimal submanifolds in Heisenberg groups. Some of the tools we develop for proving our main result are worth mentioning; in fact, we prove an extension result for intrinsic Lipschitz graphs as well as the fact that they can be uniformly approximated by smooth graphs. Both results stem from what can be considered as another contribution of the present article; that is, a new definition of intrinsic Lipschitz graphs that is equivalent to the original one, introduced by B. Franchi, R. Serapioni and F. Serra Cassano and now widely accepted. Recall, in fact, that intrinsic Lipschitz graphs in Heisenberg groups played a fundamental role in the recent proof by A. Naor and R. Young [Reference Naor and Young79] of the ‘vertical versus horizontal’ isoperimetric inequality in ${\mathbb H}^{n}$ that settled the longstanding question of determining the approximation ratio of the Goemans–Linial algorithm for the sparsest cut problem. Let us also say that our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups; a key result is for us (a version of) the celebrated constancy theorem [Reference Federer42, Reference Simon90, Reference Krantz and Parks65]. From the technical point of view, the use of currents constitutes the hardest part of the article; in fact, currents in Heisenberg groups are defined in terms of the complex of differential forms introduced by M. Rumin in [Reference Rumin85, Reference Rumin86], which is not easy to handle. Among other things, we had to provide a convenient basis of Rumin’s covectors that could be fruitfully employed in the computation of Rumin’s exterior derivatives. We were surprised by the fact that the use of standard Young tableaux from combinatorics proved to be crucial in performing this task.

It is time to introduce and discuss our results more appropriately.

1.1 Heisenberg groups and intrinsic graphs

For $n\geq 1$ , the Heisenberg group ${\mathbb H}^{n}$ is the connected, simply connected and nilpotent Lie group associated with the Lie algebra ${\mathfrak h}$ with $2n+1$ generators $X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n},T$ ; all Lie brackets between these generators are null except for

$$ \begin{align*} [X_{j},Y_{j}]=T\quad\text{for every }j=1,\dots,n. \end{align*} $$

The algebra ${\mathfrak h}$ is stratified, as it can be decomposed as ${\mathfrak h}={\mathfrak h}_{1}\oplus {\mathfrak h}_{2}$ with ${\mathfrak h}_{1}:=\operatorname *{\mathrm {span}}\{X_{j},Y_{j}:j=1,\dots ,n\}$ and ${\mathfrak h}_{2}:=\operatorname *{\mathrm {span}}\{T\}$ . The first layer ${\mathfrak h}_{1}$ in the stratification is called horizontal.

It will often be convenient to identify ${\mathbb H}^{n}$ with ${\mathbb R}^{2n+1}$ by exponential coordinates

$$ \begin{align*} {\mathbb R}^{n}\times{\mathbb R}^{n}\times{\mathbb R}\ni (x,y,t)\longleftrightarrow \exp(x_{1}X_{1}+\dots +x_{n}X_{n}+y_{1}Y_{1}+\dots+y_{n}Y_{n}+tT)\in{\mathbb H}^{n}, \end{align*} $$

where $\exp :{\mathfrak h}\to {\mathbb H}^{n}$ is the exponential map and 0 is the group identity. The Heisenberg group is a homogeneous group according to [Reference Folland and Stein43]; indeed, for $\lambda>0$ the maps $\delta _{\lambda }(x,y,t):=(\lambda x,\lambda y,\lambda ^{2} t)$ determine a one-parameter family of group automorphisms of ${\mathbb H}^{n}$ called dilations. We endow ${\mathbb H}^{n}$ with a left-invariant and homogeneous distance d, so that

$$ \begin{align*} d(qp,qp^{\prime})=d(p,p^{\prime})\quad\text{and}\quad d(\delta_{\lambda} p,\delta_{\lambda} q)=\lambda d(p,q)\qquad\forall\: p,p^{\prime},q\in{\mathbb H}^{n},\lambda>0. \end{align*} $$

It will be convenient to assume that d is rotationally invariant; that is, that

$$ \begin{align*} \|(x,y,t)\|_{\mathbb H}=\|(x^{\prime},y^{\prime},t)\|_{\mathbb H}\qquad\text{whenever }|(x,y)|=|(x^{\prime},y^{\prime})|, \end{align*} $$

where we set $\|p\|_{\mathbb H}:=d(0,p)$ for every $p\in {\mathbb H}^{n}$ . Relevant examples of rotationally invariant distances are the well-known Carnot–Carathéodory and Korányi (or Cygan–Korányi) distances.

An intensive search for a robust intrinsic notion of $C^{1}$ or Lipschitz regularity for submanifolds was conducted in the last two decades; in fact (see [Reference Ambrosio and Kirchheim3]), the Heisenberg group ${\mathbb H}^{1}$ is purely k-unrectifiable, in the sense of [Reference Federer42], for $k=2,3,4$ . It can, however, be stated that the theory of ${\mathbb H}$ -regular submanifolds (i.e., submanifolds with intrinsic $C^{1}$ regularity) is well-established; see, for instance, the beautiful paper [Reference Franchi, Serapioni and Serra Cassano51]. It turns out that ${\mathbb H}$ -regular submanifolds in ${\mathbb H}^{n}$ of low dimension $k\in \{1,\dots ,n\}$ are k-dimensional submanifolds of class $C^{1}$ (in the Euclidean sense) that are tangent to the horizontal bundle ${\mathfrak h}_{1}$ . On the contrary, ${\mathbb H}$ -regular submanifolds of low codimension $k\in \{1,\dots ,n\}$ are more complicated: they are (locally) noncritical level sets of ${\mathbb R}^{k}$ -valued maps on ${\mathbb H}^{n}$ with continuous horizontal derivatives (see Subsection 4.4 for precise definitions) and, as a matter of fact, they can have fractal Euclidean dimension [Reference Kirchheim and Serra Cassano63].

A key tool for the study of ${\mathbb H}$ -regular submanifolds is provided by intrinsic graphs. Assume that ${\mathbb V},{\mathbb W}$ are homogeneous complementary subgroups of ${\mathbb H}^{n}$ ; that is, that they are invariant under dilations, ${\mathbb V}\cap {\mathbb W}=\{0\}$ and ${\mathbb H}^{n}={\mathbb W}{\mathbb V}={\mathbb V}{\mathbb W}$ ; given $A\subset {\mathbb W}$ and a map $\phi :A\to {\mathbb V}$ , the intrinsic graph of $\phi $ is $\mathrm {gr}_{\phi }:=\{w\phi (w):w\in A\}\subset {\mathbb H}^{n}$ . It is worth recalling that, if ${\mathbb V},{\mathbb W}$ are homogeneous and complementary subgroups, then one of the two is necessarily horizontal (i.e., contained in $\exp ({\mathfrak h}_{1})$ ), abelian and of dimension $k\leq n$ , while the other has dimension $2n+1-k\geq n+1$ , is normal and contains the group center $\exp ({\mathfrak h}_{2})$ ; see [Reference Franchi, Serapioni and Serra Cassano51, Remark 3.12]. The first appearance of intrinsic graphs is most likely to be attributed to the implicit function theorem of the fundamental paper [Reference Franchi, Serapioni and Serra Cassano48], where the authors prove an ${\mathbb H}$ -rectifiability result for (boundaries of) sets with finite perimeter in ${\mathbb H}^{n}$ . As a matter of fact, ${\mathbb H}$ -regular submanifolds are locally intrinsic graphs whose properties have been studied in many papers (see, e.g., [Reference Ambrosio, Serra Cassano and Vittone4, Reference Antonelli, Di Donato and Don5, Reference Antonelli, Di Donato, Don and Le Donne6, Reference Arena and Serapioni9, Reference Bigolin and Serra Cassano17, Reference Bigolin and Serra Cassano18, Reference Citti and Manfredini29, Reference Corni32, Reference Corni and Magnani33, Reference Di Donato36, Reference Di Donato35, Reference Di Donato, Fässler and Orponen37, Reference Franchi, Serapioni and Serra Cassano51, Reference Julia, Nicolussi Golo and Vittone61, Reference Magnani70, Reference Monti and Vittone76]).

Intrinsic graphs also provide the language for introducing a theory of Lipschitz submanifolds in ${\mathbb H}^{n}$ . Observe that, while for the case of low-dimensional submanifolds one could simply consider Euclidean Lipschitz submanifolds that are almost everywhere (a.e.) tangent to the horizontal distribution, for submanifolds of low codimension there is no immediate way of modifying the ‘level set definition’ of ${\mathbb H}$ -regularity into a Lipschitz one. Intrinsic Lipschitz graphs in ${\mathbb H}^{n}$ first appeared in [Reference Franchi, Serapioni and Serra Cassano50]; their definition is stated in terms of a suitable cone property. Given $\alpha>0$ , consider the homogeneous cone of axis ${\mathbb V}$ and aperture $\alpha $ ,

$$ \begin{align*} {\mathscr C}_{\alpha}:=\{wv\in{\mathbb H}^{n}:w\in{\mathbb W},v\in{\mathbb V},\|w\|_{\mathbb H}\leq\alpha\|v\|_{\mathbb H}\}. \end{align*} $$

We say that a map $\phi :A\subset {\mathbb W}\to {\mathbb V}$ is intrinsic Lipschitz if there exists $\alpha>0$ such that

$$ \begin{align*} \mathrm{gr}_{\phi} \cap (p {\mathscr C}_{\alpha})=\{p\}\qquad\text{for every }p\in\mathrm{gr}_{\phi}. \end{align*} $$

Intrinsic Lipschitz graphs can be introduced in the more general framework of Carnot groups: apart from the elementary basics contained in Section 2, we refer to [Reference Franchi and Serapioni53] for a beautiful introduction to the topic and to [Reference Bigolin, Caravenna and Serra Cassano16, Reference Chousionis, Fässler and Orponen27, Reference Citti, Manfredini, Pinamonti and Serra Cassano30, Reference Di Donato, Fässler and Orponen37, Reference Don, Le Donne, Moisala and Vittone38, Reference Fässler, Orponen and Rigot41, Reference Franchi, Marchi and Serapioni44, Reference Franchi, Penso and Serapioni45, Reference Nicolussi Golo and Serra Cassano80, Reference Serapioni87, Reference Serra Cassano88] for several facets of the theory.

1.2 Rademacher’s theorem for intrinsic Lipschitz graphs and consequences

One of the main questions about intrinsic Lipschitz graphs concerns their almost everywhere ‘intrinsic’ differentiability. Consider an intrinsic Lipschitz map $\phi :A\to {\mathbb V}$ defined on some relatively open subset $A\subset {\mathbb W}$ . If ${\mathbb W}$ has low dimension $k\leq n$ , then (see [Reference Antonelli and Merlo8] or also [Reference Franchi, Serapioni and Serra Cassano50, Remark 3.11], [Reference Franchi and Serapioni53, Proposition 3.7]) $\mathrm {gr}_{\phi }$ is a k-dimensional submanifold with Euclidean Lipschitz regularity that is a.e. tangent to the horizontal bundle ${\mathfrak h}_{1}$ ; therefore, the problem reduces to the case of ${\mathbb H}$ -regular graphs with low codimension $k=\dim {\mathbb V}\leq n$ . A positive answer ([Reference Franchi, Serapioni and Serra Cassano52]) is known only for the case of codimension $k=1$ ; in fact, in this case $\mathrm {gr}_{\phi }$ is (part of) the boundary of a set with finite ${\mathbb H}$ -perimeter ([Reference Capogna, Danielli and Garofalo25, Reference Franchi, Serapioni and Serra Cassano46]) in ${\mathbb H}^{n}$ and one can use the rectifiability result [Reference Franchi, Serapioni and Serra Cassano48] available for such sets. A Rademacher-type theorem for intrinsic Lipschitz functions of codimension 1 was proved in Carnot groups of type $\star $ ; see [Reference Franchi, Marchi and Serapioni44]. After a preliminary version of the present article was made public, it was found that the Rademacher theorem may dramatically fail for intrinsic Lipschitz graphs of codimension 2 (or higher) even in certain Carnot groups of step 2; see [Reference Julia, Nicolussi Golo and Vittone60]. In this article, we provide a full solution to the problem in ${\mathbb H}^{n}$ , as stated in our main result.

Theorem 1.1. If $A\subset {\mathbb W}$ is open and $\phi :A\to {\mathbb V}$ is intrinsic Lipschitz, then $\phi $ is intrinsically differentiable at almost every point of A.

In Theorem 1.1, ‘almost every’ must be understood with respect to a Haar measure on the subgroup ${\mathbb W}$ ; for instance, the Hausdorff measure of dimension $2n+2-k$ . Concerning the notion of intrinsic differentiability (see Subsection 4.2), recall that left-translations and dilations of intrinsic Lipschitz graphs are intrinsic Lipschitz graphs; in particular, for every $w\in A$ and every $\lambda>0$ there exists an intrinsic Lipschitz $\phi _{w}^{\lambda }:B\to {\mathbb V}$ , defined on some open subset $B\subset {\mathbb W}$ , such that

$$ \begin{align*} \delta_{\lambda}((w\phi(w))^{-1}\mathrm{gr}_{\phi})=\mathrm{gr}_{\phi_{w}^{\lambda}}. \end{align*} $$

One then says ([Reference Arena and Serapioni9, §3.3]) that $\phi $ is intrinsically differentiable at w if, as $\lambda \to +\infty $ , the blow-ups $\phi _{w}^{\lambda }$ converge locally uniformly on ${\mathbb W}$ to an intrinsic linear map; that is, to a map $\psi :{\mathbb W}\to {\mathbb V}$ such that $\mathrm {gr}_{\psi }$ is a homogeneous subgroup of ${\mathbb H}^{n}$ with codimension k. This subgroup, which is necessarily vertical (i.e., it contains the center of ${\mathbb H}^{n}$ ) and normal, is called tangent subgroup to $\mathrm {gr}_{\phi }$ at $w\phi (w)$ and is denoted by ${\mathrm {Tan}}^{\mathbb H}_{\mathrm {gr}_{\phi }}(w\phi (w))$ .

For the reader’s convenience, the proof of Theorem 1.1 is sketched at the end of the Introduction. We are now going to introduce a few consequences of our main result: the first one is a Lusin-type theorem for intrinsic Lipschitz graphs.

Theorem 1.2. Let $A\subset {\mathbb W}$ be an open set and $\phi :A\to {\mathbb V}$ an intrinsic Lipschitz function. Then for every $\varepsilon>0$ there exists an intrinsic Lipschitz function $\psi :A\to {\mathbb V}$ such that $\mathrm {gr}_{\psi }$ is a ${\mathbb H}$ -regular submanifold and

$$ \begin{align*} {\mathscr S}^{Q-k}((\mathrm{gr}_{\phi}\,\Delta\, \mathrm{gr}_{\psi})\cup\{p\in\mathrm{gr}_{\phi}\cap \mathrm{gr}_{\psi}:{\mathrm{Tan}}^{\mathbb H}_{\mathrm{gr}_{\phi}}(p)\neq {\mathrm{Tan}}^{\mathbb H}_{\mathrm{gr}_{\psi}}(p)\})<\varepsilon. \end{align*} $$

As is customary, the integer $Q:=2n+2$ denotes the homogeneous dimension of ${\mathbb H}^{n}$ and ${\mathscr S}^{Q-k}$ is the spherical Hausdorff measure of dimension $Q-k$ ; by $A\Delta B:=(A\setminus B)\cup (B\setminus A)$ we denote the symmetric difference of sets $A,B$ . Theorem 1.2 is part of Theorem 7.2; the latter stems from the equivalent definition of intrinsic Lipschitz graphs provided by Theorem 1.4 and is proved by an adaptation of the classical argument of Whitney’s extension theorem; see also [Reference Franchi, Serapioni and Serra Cassano48, Reference Franchi, Serapioni and Serra Cassano49, Reference Vittone92, Reference Vodop’yanov and Pupyshev94]. Theorem 1.2 implies that, as in the Euclidean case, the notion of ${\mathbb H}$ -rectifiability (Definition 4.22) can be equivalently defined in terms of either ${\mathbb H}$ -regular submanifolds or intrinsic Lipschitz graphs; see Corollary 7.4. We refer to [Reference Antonelli and Le Donne7, Reference Bigolin and Vittone19, Reference Chousionis, Fässler and Orponen27, Reference Chousionis, Magnani and Tyson28, Reference Cole and Pauls31, Reference Di Donato, Fässler and Orponen37, Reference Fässler, Orponen and Rigot41, Reference Mattila, Serapioni and Serra Cassano73, Reference Merlo74, Reference Merlo75] for more about rectifiability in Heisenberg groups.

We stress the fact that Rademacher’s Theorem 1.1 also allows defining a canonical current carried by the graph of an intrinsic Lipschitz map $\phi :{\mathbb W}\to {\mathbb V}$ . This current turns out to have zero boundary; see Proposition 7.5.

A further consequence of Theorem 1.2 is an area formula for intrinsic Lipschitz graphs of low codimension. For ${\mathbb H}$ -regular submanifolds, area formulae are proved in [Reference Franchi, Serapioni and Serra Cassano48, Reference Franchi, Serapioni and Serra Cassano51, Reference Ambrosio, Serra Cassano and Vittone4] for submanifolds of codimension 1 and in [Reference Corni and Magnani33] for higher codimension (see also [Reference Magnani71]). For intrinsic Lipschitz graphs of low dimension, an area formula is proved in [Reference Antonelli and Merlo8, Theorem 1.1]. Our area formula is stated in Theorem 1.3 and, once Lusin’s Theorem 1.2 is available, it is a quite simple consequence of [Reference Corni and Magnani33, Theorem 1.2], where a similar area formula is proved for intrinsic graphs that are also ${\mathbb H}$ -regular submanifolds. As in [Reference Corni and Magnani33], the symbol $J^{\phi }\phi (w)$ denotes the intrinsic Jacobian of $\phi $ at w (see Definition 4.9), while $C_{n,k}$ denotes a positive constant, depending only on $n,k$ and the distance d, which will be introduced later in Proposition 1.9.

Theorem 1.3. Assume that the subgroups ${\mathbb W},{\mathbb V}$ are orthogonalFootnote 1 and let $\phi :A\to {\mathbb V}$ be an intrinsic Lipschitz map defined on some Borel subset $A\subset {\mathbb W}$ ; then for every Borel function $h:\mathrm {gr}_{\phi }\to [0,+\infty )$ , there holds

$$ \begin{align*} \int_{\mathrm{gr}_{\phi}} h \:d{\mathscr S}^{Q-k} =C_{n,k}\int_{A} (h\circ\Phi)J^{\phi}\phi\:d{\mathscr L}^{2n+1-k}, \end{align*} $$

where $\Phi $ denotes the graph map $\Phi (w):=w\phi (w).$

By abuse of notation, ${\mathscr L}^{2n+1-k}$ denotes the Haar measure on ${\mathbb W}$ associated with the canonical identification of ${\mathbb W}$ with ${\mathbb R}^{2n+1-k}$ induced by exponential coordinates. It is worth observing that Theorem 1.3 and Proposition 1.9 are the only points where we use the rotational invariance of the distance d. In case of general distances, area formulae for intrinsic Lipschitz graphs can be easily deduced using Theorem 1.2 and [Reference Corni and Magnani33, Theorem 1.2], but they are slightly more complicated than ours, as they involve a certain area factor that depends on the tangent plane to the graph.

1.3 Equivalent definition, extension and approximation of intrinsic Lipschitz graphs

We now introduce two of the ingredients needed in the proof of Theorem 1.1 that are of independent interest: namely, an extension theorem for intrinsic Lipschitz graphs in the spirit of the classical McShane–Whitney theorem and an approximation result by means of smooth graphs. They are stated in Theorems 1.5 and 1.6 and are both based on a new, equivalent definition of intrinsic Lipschitz graphs (Theorem 1.4), which can be regarded as another contribution of this article.

Our alternative definition of intrinsic Lipschitz graphs appeared in [Reference Vittone93] for graphs of codimension 1; it can be seen as a generalisation of the original level-set definition of ${\mathbb H}$ -regular submanifolds. Observe, however, that it is not immediate to give a level-set definition even for Lipschitz submanifolds of codimension 1 in ${\mathbb R}^{n}$ ; in fact, every closed set $S\subset {\mathbb R}^{n}$ is the level set of some Lipschitz function; for instance, the distance from S. Anyway, we leave as an exercise to the reader the following observation, which was actually the starting point of [Reference Vittone93]: a set $S\subset {\mathbb R}^{n}={\mathbb R}^{n-1}\times {\mathbb R}$ is (contained in) the graph of a Lipschitz function $\phi :{\mathbb R}^{n-1}\to {\mathbb R}$ if and only if there exist $\delta>0$ and a Lipschitz function $f:{\mathbb R}^{n}\to {\mathbb R}$ such that $S\subset \{x\in {\mathbb R}^{n}:f(x)=0\}$ and $\frac {\partial f}{\partial {x_{n}}}\geq \delta $ a.e. on ${\mathbb R}^{n}$ .

Since their proofs present no extra difficulty with respect to the Heisenberg case, Theorems 1.4, 1.5 and 1.6 are stated in the more general setting of a Carnot group ${\mathbb G}$ where two homogeneous complementary subgroups ${\mathbb W},{\mathbb V}$ are fixed with ${\mathbb V}$ horizontal. This means that ${\mathbb V}\subset \exp ({\mathfrak g}_{1})$ , where ${\mathfrak g}_{1}$ is the first layer in the stratification of the Lie algebra of ${\mathbb G}$ . When ${\mathbb V}$ is horizontal, we say that an intrinsic Lipschitz graph $\phi :{\mathbb W}\to {\mathbb V}$ is co-horizontal; see [Reference Antonelli, Di Donato, Don and Le Donne6]. Observe that ${\mathbb V}$ is necessarily abelian and there exists a homogeneous isomorphism according to which we can identify ${\mathbb V}$ with ${\mathbb R}^{k}$ ; see (2.5). This identification is understood in the scalar product appearing in (1.2).

Theorem 1.4. Assume that a splitting ${\mathbb G}={\mathbb W}{\mathbb V}$ is fixed in such a way that the subgroup ${\mathbb V}$ is horizontal; set $k:=\dim {\mathbb V}$ . If $S\subset {\mathbb G}$ is not empty, then the following statements are equivalent:

  1. (a) there exist $A\subset {\mathbb W}$ and an intrinsic Lipschitz map $\phi :A\to {\mathbb V}$ such that $S=\mathrm {gr}_{\phi }$ ;

  2. (b) there exist $\delta>0$ and a Lipschitz map $f:{\mathbb G}\to {\mathbb R}^{k}$ such that

    (1.1) $$ \begin{align} & S\subset \{x\in{\mathbb G}:f(x)=0\} \end{align} $$
    (1.2) $$ \begin{align} \text{and}\quad& \langle f(xv)-f(x),v\rangle \geq \delta|v|^{2}\qquad\text{for every }v\in{\mathbb V}\text{ and }x\in{\mathbb G}. \end{align} $$

It is worth remarking that, if $X_{1},\dots , X_{k}\in {\mathfrak g}_{1}$ are such that ${\mathbb V}=\exp (\operatorname *{\mathrm {span}}\{X_{1},\dots ,X_{k}\})$ , then statement (1.2) is equivalent to the a.e. uniform ellipticity (a.k.a. coercivity) of the matrix col $[\,X_{1}f(x)\,|\,\dots \,|\,X_{k} f(x)\,]$ ; see Remark 2.7. In case $k=1$ , Theorem 1.4 was proved in [Reference Vittone93, Theorem 3.2].

Let us underline two of the most interesting features of this alternative definition. First, it allows for a definition of co-horizontal intrinsic Lipschitz submanifolds in the more general setting of Carnot–Carathéodory spaces, as in [Reference Vittone93]. Second, it gives gratis an extension result for intrinsic Lipschitz maps; in fact (Remark 2.8), the level set $\{x\in {\mathbb G}:f(x)=0\}$ appearing in (1.1) is the graph of an intrinsic Lipschitz map that is defined on the whole ${\mathbb W}$ and extends $\phi $ . We can then state the following result.

Theorem 1.5. Let $A\subset {\mathbb W}$ and $\phi :A\to {\mathbb V}$ be a co-horizontal intrinsic Lipschitz graph; then there exists an intrinsic Lipschitz extension $\tilde {\phi }:{\mathbb W}\to {\mathbb V}$ of $\phi $ . Moreover, $\tilde {\phi }$ can be chosen in such a way that its intrinsic Lipschitz constant is controlled in terms of the intrinsic Lipschitz constant of $\phi $ .

Theorem 1.5 was proved in [Reference Vittone93, Proposition 3.4] for the case of codimension $k=1$ ; see also [Reference Franchi, Serapioni and Serra Cassano52, Reference Franchi and Serapioni53, Reference Naor and Young79, Reference Rigot84].

In Proposition 2.10 we use a standard approximation argument based on group convolutions (see, e.g., [Reference Folland and Stein43, §1.B]) to show that the function f appearing in Theorem 1.4 can be chosen with the additional property that $f\in C^{\infty }(\{x\in {\mathbb G}:f(x)\neq 0\})$ . This fact has the following consequence.

Theorem 1.6. Let $A\subset {\mathbb W}$ and $\phi :A\to {\mathbb V}$ be a co-horizontal intrinsic Lipschitz graph. Then there exists a sequence $(\phi _{i})_{i\in \mathbb N}$ of $C^{\infty }$ -regular and intrinsic Lipschitz maps $\phi _{i}:{\mathbb W}\to {\mathbb V}$ such that

$$ \begin{align*} \phi_{i}\to\phi\text{ uniformly in }A\text{ as }i\to\infty\,. \end{align*} $$

Moreover, the intrinsic Lipschitz constant of $\phi _{i}$ is bounded, uniformly in i, in terms of the intrinsic Lipschitz constant of $\phi $ .

A similar result has been proved in [Reference Citti, Manfredini, Pinamonti and Serra Cassano30] for intrinsic Lipschitz graphs of codimension 1 in Heisenberg groups; see also [Reference Ambrosio, Serra Cassano and Vittone4, Reference Antonelli, Di Donato, Don and Le Donne6, Reference Monti and Vittone76, Reference Vittone93].

1.4 Currents and the constancy theorem

As in the classical setting, currents in Heisenberg groups are defined in duality with spaces of smooth forms with compact support; here, however, the De Rham complex must be replaced by the complex introduced by M. Rumin [Reference Rumin85, Reference Rumin86] in the setting of contact manifolds. The construction of the spaces $\Omega _{\mathbb H}^{m}$ of Heisenberg differential m -forms is detailed in Subsection 3.2; here we only recall that, for $1\leq k\leq n$ , Heisenberg forms of codimension k are smooth functions on ${\mathbb H}^{n}$ with values in a certain subspace ${\mathcal J}^{2n+1-k}$ of $(2n+1-k)$ -covectors. We denote by ${\mathcal J}_{2n+1-k}$ the (formal) dual of ${\mathcal J}^{2n+1-k}$ ; clearly, every $(2n+1-k)$ -vector t canonically induces an element $[t]_{{\mathcal J}}\in {\mathcal J}_{2n+1-k}$ defined by $[t]_{{\mathcal J}}(\lambda ):=\langle \, t\mid \lambda \,\rangle $ , where $\langle \,\cdot \mid \cdot \,\rangle $ is the standard pairing vectors-covectors. See Subsection 3.1 and Subsection 3.2 for more details.

The starting point of the theory of Heisenberg currents is the existence of a linear second-order operator $D:\Omega _{\mathbb H}^{n}\to \Omega _{\mathbb H}^{n+1}$ such that the sequence

$$ \begin{align*} 0\to{\mathbb R}\to\Omega_{\mathbb H}^{0}\stackrel{d}{\to}\Omega_{\mathbb H}^{1}\stackrel{d}{\to}\dots\stackrel{d}{\to}\Omega_{\mathbb H}^{n}\stackrel{D}{\to}\Omega_{\mathbb H}^{n+1}\stackrel{d}{\to} \dots \stackrel{d}{\to}\Omega_{\mathbb H}^{2n+1}\to 0 \end{align*} $$

is exact, where d is (the operator induced by) the standard exterior derivative. A Heisenberg m-current ${\mathsf T}$ is by definition a continuous linear functional on the space ${\mathcal D}_{\mathbb H}^{m}\subset \Omega _{\mathbb H}^{m}$ of Heisenberg m-forms with compact support. The boundary $\partial {\mathsf T}$ of ${\mathsf T}$ is the Heisenberg $(m-1)$ -current defined, for every $\omega \in {\mathcal D}_{\mathbb H}^{m-1}$ , by

$$ \begin{align*} \begin{array}{ll} \partial {\mathsf T}(\omega):={\mathsf T}(d\omega)&\quad\text{if }m\neq n+1\\ \partial {\mathsf T}(\omega):={\mathsf T}(D\omega)&\quad\text{if }m= n+1. \end{array} \end{align*} $$

We say that ${\mathsf T}$ is locally normal if both ${\mathsf T}$ and $\partial {\mathsf T}$ have locally finite mass; that is, if they have order 0 in the sense of distributions. Recall that, if ${\mathsf T}$ has locally finite mass, then there exist a Radon measure $\mu $ on ${\mathbb H}^{n}$ and a locally $\mu $ -integrable function $\tau $ with values in a suitable space of multivectors (which, for $m\geq n+1$ , is precisely ${\mathcal J}_{m}$ ) such that ${\mathsf T}=\tau \mu $ , where

$$ \begin{align*} \tau\mu(\omega):=\int_{{\mathbb H}^{n}}\langle\,\tau(p)\mid\omega(p)\,\rangle\,d\mu(p)\qquad\text{for every }\omega\in{\mathcal D}_{\mathbb H}^{m}. \end{align*} $$

One can also assume that $|\tau |=1\ \mu $ -a.e., where $|\cdot |$ denotes some fixed norm on multivectorsFootnote 2 ; in this case, we write $\!\vec {\,{\mathsf T}}$ and $\|{\mathsf T}\|$ in place of $\tau $ and $\mu $ , respectively.

Relevant examples of currents will be for us those concentrated on ${\mathbb H}$ -rectifiable sets of low codimension. Recall that a set $R\subset {\mathbb H}^{n}$ is locally ${\mathbb H}$ -rectifiable of codimension $k\in \{1,\dots ,n\}$ if is locally finite and R can be covered by countably many ${\mathbb H}$ -regular submanifolds of codimension k plus a ${\mathscr S}^{Q-k}$ -negligible set. In this case, a (unit) approximate tangent $(2n+1-k)$ -vector $t^{\mathbb H}_{R}(p)$ to R can be defined at ${\mathscr S}^{Q-k}$ -a.e. $p\in {\mathbb R}$ ; see Subsection 4.4. We denote by the Heisenberg current naturally associated with R.

A fundamental result in the classical theory of currents is the constancy theorem (see, e.g., [Reference Federer42, 4.1.7] and [Reference Simon90, Theorem 26.27]), which states that, if ${\mathsf T}$ is an n-dimensional current in ${\mathbb R}^{n}$ such that $\partial {\mathsf T}=0$ , then ${\mathsf T}$ is constant; that is, there exists $c\in {\mathbb R}$ such that ${\mathsf T}(\omega )=c\int _{{\mathbb R}^{n}}\omega $ for every smooth n-form $\omega $ with compact support. A more general version of the constancy theorem can be proved for currents supported on an m-dimensional plane ${\mathscr P}\subset {\mathbb R}^{n}$ : if ${\mathsf T}$ is an m-current with support in ${\mathscr P}$ and such that $\partial {\mathsf T}=0$ , then there exists $c\in {\mathbb R}$ such that ${\mathsf T}(\omega )=c\int _{{\mathscr P}}\omega $ for every smooth m-form $\omega $ with compact support. For this statement, see, for example, [Reference Krantz and Parks65, Proposition 7.3.5]. The following Theorem 1.7 can be considered as the Heisenberg analogue of this more general constancy theorem; besides its importance for the present article, Theorem 1.7 is a fundamental tool for the outcomes of the recent [Reference Julia, Nicolussi Golo and Vittone62].

Theorem 1.7. Let $k\in \{1,\dots ,n\}$ be fixed and let ${\mathsf T}$ be a Heisenberg $(2n+1-k)$ -current supported on a vertical plane ${\mathscr P}\subset {\mathbb H}^{n}$ of dimension $2n+1-k$ . Assume that $\partial {\mathsf T}=0$ ; then there exists a constant $c\in {\mathbb R}$ such that .

Using a procedure involving projection on planes (see [Reference Šilhavý89, Theorem 4.2]; let us mention also [Reference Alberti and Marchese1, §5] and [Reference Alberti, Massaccesi and Stepanov2] for some related results), the (version on planes of the) constancy theorem in ${\mathbb R}^{n}$ has the following consequence: if $R\subset {\mathbb R}^{n}$ is an m-rectifiable set and ${\mathsf T}=\tau \mu $ is a normal m-current, where $\mu $ is a Radon measure and $\tau $ is a locally $\mu $ -integrable m-vectorfield with $\tau \neq 0 \ \mu $ -a.e., then

  1. (i) is absolutely continuous with respect to the Hausdorff measure and

  2. (ii) $\tau $ is tangent to R at $\mu $ -almost every point of R.

A consequence of this fact, which might help explaining its geometric meaning, is the following one: if ${\mathsf T}=\tau \mu $ is a normal current concentratedFootnote 3 on a rectifiable set R, then and $\tau $ is necessarily tangent to $R \ \mu $ -almost everywhere.Footnote 4

In our proof of Rademacher’s Theorem 1.1, we will utilise the following result, which is the Heisenberg counterpart of the ‘tangency’ property (ii) above; we were not able to deduce any ‘absolute continuity’ statement analogous to (i) because no good notion of projection on planes is available in ${\mathbb H}^{n}$ . Notice, however, that, in the special case when ${\mathsf T}$ is concentrated on a vertical plane, Theorem 1.7 allows deducing a complete result including absolute continuity.

Theorem 1.8. Let $k\in \{1,\dots ,n\}$ and let a locally normal Heisenberg $(2n+1-k)$ -current ${\mathsf T}$ and a locally ${\mathbb H}$ -rectifiable set $R\subset {\mathbb H}^{n}$ of codimension k be fixed. Then

$$ \begin{align*} \!\vec{\,{\mathsf T}}(p)\text{ is a multiple of }[t^{\mathbb H}_{R}(p)]_{{\mathcal J}}\qquad \text{for }\|{\mathsf T}\|_{a}\text{-a.e. }p. \end{align*} $$

In Theorem 1.8 we decomposed $\|{\mathsf T}\|=\|{\mathsf T}\|_{a}+\|{\mathsf T}\|_{s}$ as the sum of the absolutely continuous and singular part of $\|{\mathsf T}\|$ with respect to . Observe that $t^{\mathbb H}_{R}$ is defined only ${\mathscr S}^{Q-k}$ -almost everywhere on R; hence, it could be undefined on a set with positive $\|{\mathsf T}\|_{s}$ -measure. The geometric content of Theorem 1.8 is again clear: for a current ${\mathsf T}$ concentrated on R to be normal, it is necessary that $\!\vec {\,{\mathsf T}}$ is almost everywhere tangent to R.

The proof of Theorem 1.8 follows a blow-up strategy according to which one can prove that, at ${\mathscr S}^{Q-k}$ -a.e. $p\in R$ , the current has zero boundary, where ${\mathrm {Tan}}^{\mathbb H}_{R}(p)=\exp (\operatorname *{\mathrm {span}} t^{\mathbb H}_{R}(p))$ is the approximate tangent plane to R at p. Proposition 5.3 shows that this is possible only if $\!\vec {\,{\mathsf T}}(p)$ is a multiple of $[t^{\mathbb H}_{R}(p)]_{{\mathcal J}}$ . Proposition 5.3 is essentially a simpler version of Theorem 1.7; its classical counterpart can be found, for instance, in [Reference Giaquinta, Modica and Souček57, Lemma 1 in §3.3.2]. The proof of Proposition 5.3 Footnote 5 consists in feeding the given boundaryless current with (the differential of) enough test forms in order to eventually deduce the desired ‘tangency’ property. Apart from the computational difficulties pertaining to the second-order operator D (at least in case $k=n$ ), one demanding task we had to face was the search for a convenient basis of ${\mathcal J}^{2n+1-k}$ ; see Subsection 1.5.

We conclude this section with an important observation. Assume that S is an oriented submanifold of codimension k that is (Euclidean) $C^{1}$ -regular; in particular, the tangent vector $t^{\mathbb H}_{S}$ is defined except at characteristic points of S, which, however, are ${\mathscr S}^{Q-k}$ -negligible [Reference Balogh15, Reference Magnani69]. Then, on the one side, S induces the natural Heisenberg current ; on the other side, associated to S is also the classical current defined by for every $(2n+1-k)$ -form $\omega $ with compact support. The following fact holds true provided the homogeneous distance d is rotationally invariant.

Proposition 1.9. Let $k\in \{1,\dots ,n\}$ ; then there exists a positive constant $C_{n,k}$ , depending on $n,k$ and the rotationally invariant distance d, such that for every $C^{1}$ -regular submanifold $S\subset {\mathbb H}^{n}$ of codimension k,


In particular, if S is a submanifold without boundary, then is a Heisenberg $(2n-k)$ -current.

In other words, and coincide, as Heisenberg currents, up to a multiplicative constant. This is remarkable. The first part of the statement of Proposition 1.9 is proved in Lemma 3.31, while the second one is a consequence of the fact that the operator D is the composition of the differential d with another operator; see Corollary 3.34. For the exact value of $C_{n,k}$ , see Remark 4.21. Proposition 1.9 is crucial in the proof of our main result Theorem 1.1.

1.5 A basis for Rumin’s spaces ${\mathcal J}^{2n+1-k}$

We believe it is worth introducing, at least quickly, the basis of ${\mathcal J}^{2n+1-k}$ that we use; we need some preliminary notation. Assume that the elements of a finite subset $M\subset \mathbb N$ with cardinality $|M|=m$ are arranged (each element of M appearing exactly once) in a tableau with two rows, the first row displaying $\ell \geq \frac m2$ elements $R^{1}_{1},\dots ,R^{1}_{\ell }$ and the second one displaying $m-\ell \geq 0$ elements $R^{2}_{1},\dots ,R^{2}_{m-\ell }$ , as follows:

Such an R is called Young tableau (see, e.g., [Reference Fulton55]). Clearly, R has to be read as a $(2\times \ell )$ rectangular tableau when $\ell =m/2$ while, in case $\ell =m$ , we agree that the second row is empty. Given such an R, define the $2\ell $ -covector

$$ \begin{align*} \alpha_{R}:=(dxy_{R^{1}_{1}}- dxy_{R^{2}_{1}})\wedge(dxy_{R^{1}_{2}}- dxy_{R^{2}_{2}})\wedge\dots\wedge(dxy_{R^{1}_{m-\ell}}- dxy_{R^{2}_{m-\ell}})\wedge dxy_{R^{1}_{m-\ell+1}}\wedge\dots\wedge dxy_{R^{1}_{\ell}}, \end{align*} $$

where for shortness we set $dxy_{i}:=dx_{i}\wedge dy_{i}$ ; when $\ell =m$ (i.e., when the second row of R is empty), we agree that $\alpha _{R}=dxy_{R^{1}_{1}}\wedge \dots \wedge dxy_{R^{1}_{\ell }}$ . One key observation is the fact that

(1.4) $$ \begin{align} \alpha_{R}\wedge\sum_{i\in M}dxy_{i}=0, \end{align} $$

which is essentially a consequence of the equality $(dxy_{i}-dxy_{j})\wedge (dxy_{i}+dxy_{j})=0$ .

Before stating Proposition 1.10, we need some further notation. First, we say that R is a standard Young tableau when the elements in each row and each column of R are in increasing order; that is, when $R^{i}_{j}<R^{i}_{j+1}$ and $R^{1}_{j}<R^{2}_{j}$ . Second, given $I=\{i_{1},\dots ,i_{|I|}\}\subset \{1,\dots ,n\}$ with $i_{1}<i_{2}<\dots <i_{|I|}$ , we write

$$ \begin{align*} dx_{I}:=dx_{i_{1}}\wedge\dots\wedge dx_{i_{|I|}},\qquad dy_{I}:=dy_{i_{1}}\wedge\dots\wedge dy_{i_{|I|}}. \end{align*} $$

Eventually, we denote by $\theta :=dt+\frac 12\sum _{i=1}^{n}(y_{i}dx_{i}-x_{i}dy_{i})$ the contact form on ${\mathbb H}^{n}$ , which is left-invariant and then can be thought of as a covector in $\Large \wedge ^{1}{\mathfrak h}$ . Observe that $\theta $ vanishes on horizontal vectors.

Proposition 1.10. For every $k\in \{1,\dots ,n\}$ , a basis of ${\mathcal J}^{2n+1-k}$ is provided by the elements of the form $dx_{I}\wedge dy_{J}\wedge \alpha _{R}\wedge \theta $ , where $(I,J,R)$ ranges among those triples such that

  • $I\subset \{1,\dots ,n\}$ , $J\subset \{1,\dots ,n\}$ , $|I|+|J|\leq k$ and $I\cap J=\emptyset $ ;

  • R is a standard Young tableau containing the elements of $\{1,\dots ,n\}\setminus (I\cup J)$ arranged in two rows of length, respectively, $(2n-k-|I|-|J|)/2$ and $(k-|I|-|J|)/2$ .

Proposition 1.10 follows from Corollary 3.22. Observe that the tableaux R appearing in the statement are rectangular exactly in case $k=n$ . In this case, it might happen that $I\cup J=\{1,\dots ,n\}$ ; that is, that R is the empty table. If so, we agree that $\alpha _{R}=1$ . It is also worth observing that the covectors $\lambda _{I,J,R}=dx_{I}\wedge dy_{J}\wedge \alpha _{R}\wedge \theta $ appearing in Proposition 1.10 indeed belong to ${\mathcal J}^{2n+1-k}$ because $\lambda _{I,J,R}\wedge \theta =0$ (by definition) and $\lambda _{I,J,R}\wedge d\theta =0$ , which comes as a consequence of (1.4) and the fact that $d\theta =-\sum _{i=1}^{n} dxy_{i}$ is, up to a sign, the standard symplectic form.

During the preparation of this article, we became aware that a basis of ${\mathcal J}^{2n+1-k}$ is provided also in the paper [Reference Baldi, Barnabei and Franchi11]: however, the basis in [Reference Baldi, Barnabei and Franchi11] is presented by induction on n, while ours is given directly and is somewhat manageable in the computations we need.

1.6 Sketch of the proof of Rademacher’s Theorem 1.1

For the reader’s convenience, we provide a sketch of the proof of our main result. Let $\phi :A\subset {\mathbb W}\to {\mathbb V}$ be intrinsic Lipschitz; by Theorem 1.5 we can assume that $A={\mathbb W}$ . We now use Theorem 1.6 to produce a sequence of smooth maps $\phi _{i}:{\mathbb W}\to {\mathbb V}$ converging uniformly to $\phi $ : it can be easily proved that the associated Heisenberg currents converge (possibly up to a subsequence) to a current ${\mathsf T}$ supported on $\mathrm {gr}_{\phi }$ and, actually, that for some bounded function $\tau :\mathrm {gr}_{\phi }\to {\mathcal J}_{2n+1-k}\setminus \{0\}$ . Moreover, we have for every i because of Proposition 1.9; therefore, also $\partial {\mathsf T}=0$ . As we will see, this equality carries the relevant geometric information.

Our aim is to prove that, at a.e. $w\in {\mathbb W}$ , the blow-up of $\phi $ at w (i.e., the limit as $r\to +\infty $ of $\delta _{r}((w\phi (w))^{-1}\mathrm {gr}_{\phi })$ ) is the graph of an intrinsic linear map; a priori, however, there could exist many possible blow-up limits $\psi $ associated with different diverging scaling sequences $(r_{j})_{j}$ . In Lemma 4.16 we prove the following: for a.e. $\bar {w}\in {\mathbb W}$ , all of the possible blow-ups $\psi $ of $\phi $ at $\bar {w}$ are t-invariant; that is,

$$ \begin{align*} \psi(w\exp(tT))=\psi(w(0,0,t))=\psi(w)\qquad\text{for every }t\in{\mathbb R},w\in{\mathbb W}. \end{align*} $$

The proof of Lemma 4.16 makes use of Rademacher’s theorem proved for the case of codimension 1 in [Reference Franchi, Serapioni and Serra Cassano52].

Let then $\bar {w}$ be such a point and fix a t-invariant blow-up $\psi $ of $\phi $ at $\bar {w}$ associated with a scaling sequence $(r_{j})_{j}$ . It is a good point to notice that, being both intrinsic Lipschitz (because it is the limit of uniformly intrinsic Lipschitz maps) and t-invariant, $\psi $ is necessarily Euclidean Lipschitz; see Lemma 4.15. Consider now the current ${\mathsf T}_{\infty }$ defined (up to passing to a subsequence) as the blow-up limit along $(r_{j})_{j}$ of ${\mathsf T}$ at $\bar p:=\bar {w}\phi (\bar {w})\in \mathrm {gr}_{\phi }$ , namely,

$$ \begin{align*} {\mathsf T}_{\infty}:=\lim_{j\to\infty}(\delta_{r_{j}}\circ L_{\bar p^{-1}})_\#{\mathsf T}, \end{align*} $$

where $L_{\bar p^{-1}}$ denotes left-translation by $\bar p^{-1}$ and the subscript $\#$ denotes push-forward. If one assumes that $\bar p$ is also a Lebesgue point (in a suitable sense) of the function $\tau $ , then the following properties hold for ${\mathsf T}_{\infty }$ :

  • for some positive and bounded function f on $\mathrm {gr}_{\psi }$ ;

  • $\mathrm {gr}_{\psi }$ is locally Euclidean rectifiable and, in particular, it is locally ${\mathbb H}$ -rectifiable;

  • $\partial {\mathsf T}_{\infty }=0$ , because ${\mathsf T}_{\infty }$ is limit of boundaryless currents.

We can then apply Theorem 1.8 to deduce that $[t^{\mathbb H}_{\mathrm {gr}_{\psi }}(p)]_{{\mathcal J}}$ is a multiple of $\tau (\bar p)$ for a.e. $p\in \mathrm {gr}_{\psi }$ . By t-invariance, the unit tangent vector $t_{\mathrm {gr}_{\psi }}(p)$ coincides with $t^{\mathbb H}_{\mathrm {gr}_{\psi }}(p)$ . Summarising, we have a t-invariant Euclidean Lipschitz submanifold $\mathrm {gr}_{\psi }$ whose unit tangent vector $t_{\mathrm {gr}_{\psi }}$ is always vertical (i.e., of the form $t_{\mathrm {gr}_{\psi }}=t^{\prime }\wedge T$ for a suitable multivector $t^{\prime }$ ) and has the property that, for a.e. point p, $[t_{\mathrm {gr}_{\psi }}(p)]_{{\mathcal J}}$ is a multiple of $\tau (\bar p)\in {\mathcal J}_{2n+1-k}\setminus \{0\}$ . If we could guarantee that there is a unique (up to a sign) unit simple vector $\bar t$ that is vertical and such that $[\,\bar t\,]_{{\mathcal J}}$ is a multiple of $\tau (\bar p)$ , then we would conclude that $\mathrm {gr}_{\psi }$ is always tangent to that particular $\bar t$ ; that is, that $\mathrm {gr}_{\psi }$ is a vertical plane ${\mathscr P}$ . Since $\bar t$ (and then ${\mathscr P}$ ) depends only on $\bar p$ and not on the particular sequence $(r_{j})_{j}$ , the blow-up ${\mathscr P}$ is unique and is the graph of an intrinsic linear map $\psi $ : this would conclude the proof.

Unluckily, this is not always the case; in fact, in the second Heisenberg group ${\mathbb H}^{2}$ the unit simple vertical 3-vectors $X_{1}\wedge Y_{1}\wedge T$ and $-X_{2}\wedge Y_{2}\wedge T$ have the property that

$$ \begin{align*} [ X_{1}\wedge Y_{1}\wedge T]_{{\mathcal J}} = [- X_{2}\wedge Y_{2}\wedge T]_{{\mathcal J}}. \end{align*} $$

This, however, is basically the worst-case scenario. A key, technically demanding result is Proposition 3.38, where we prove that there exist at most two linearly independent unit simple vertical vectors $\bar t_{1},\bar t_{2}$ such that $[ \bar t_{1}]_{{\mathcal J}} = [\bar t_{2}]_{{\mathcal J}}$ are multiples of $\tau (\bar p)$ ; moreover, the planes ${\mathscr P}_{1},{\mathscr P}_{2}$ associated (respectively) with $\pm \bar t_{1},\pm \bar t_{2}$ are not rank 1 connected; that is, $\dim {\mathscr P}_{1}\cap {\mathscr P}_{2}$ has codimension at least 2 in ${\mathscr P}_{1}$ (equivalently, in ${\mathscr P}_{2}$ ). This means that the vertical Euclidean Lipschitz submanifold $\mathrm {gr}_{\psi }$ has at most two possible tangent planes ${\mathscr P}_{1},{\mathscr P}_{2}$ ; however (see, e.g., [Reference Ball and James14, Proposition 1] or [Reference Müller78, Proposition 2.1]), the fact that these two planes are not rank 1 connected forces $\mathrm {gr}_{\psi }$ to be a plane (either ${\mathscr P}_{1}$ or ${\mathscr P}_{2}$ ) itself.

This is not the conclusion yet: we have for the moment proved that, for a.e. $\bar {w}\in {\mathbb W}$ , all of the possible blow-ups of $\phi $ at $\bar {w}$ are either the map $\psi _{1}$ parametrising ${\mathscr P}_{1}$ or the map $\psi _{2}$ parametrising ${\mathscr P}_{2}$ ; both are determined by $\tau (\bar p)$ (i.e., by $\bar {w}$ ) only. However, it is not difficult to observe that the family of all possible blow-ups of $\phi $ at a fixed point must enjoy a suitable connectedness property; hence, it cannot consist of the two points $\psi _{1},\psi _{2}$ only. This proves the uniqueness of blow-ups and concludes the proof of our main result.

1.7 Structure of the article

In Section 2 we introduce intrinsic Lipschitz graphs in Carnot groups and prove Theorems 1.4, 1.5 and 1.6. Heisenberg groups are introduced in Section 3, where we focus on the algebraic preliminary material, in particular, about multilinear algebra and Rumin’s complex. We also provide the basis of Rumin’s spaces of Proposition 1.10, introduce Heisenberg currents and prove Proposition 1.9. Eventually, we state Proposition 3.38, which we use in the proof of Rademacher’s Theorem 1.1 and whose long and tedious proof is postponed to Appendix A. In Section 4 we deal with intrinsic Lipschitz graphs of low codimension; in particular, we define intrinsic differentiability and we prove the crucial Lemma 4.16. We also introduce ${\mathbb H}$ -regular submanifolds and ${\mathbb H}$ -rectifiable sets and we study (Euclidean) $C^{1}$ -regular intrinsic graphs. Section 5 is devoted to the proof of the constancy-type Theorems 1.7 and 1.8. The proof of Rademacher’s Theorem 1.1 is provided in Section 6. Section 7 contains the applications of our main result concerning Lusin’s Theorem 1.2, the equivalence between ${\mathbb H}$ -rectifiability and ‘Lipschitz’ ${\mathbb H}$ -rectifiability (Corollary 7.4) and the area formula of Theorem 1.3.

2 Intrinsic Lipschitz graphs in Carnot groups: extension and approximation results

In this section we introduce Carnot groups and intrinsic Lipschitz graphs; our goal is to prove the extension and approximation results stated in Theorems 1.5 and 1.6. These two results are used later in the article for intrinsic Lipschitz graphs in Heisenberg groups; however, they can be proved with no extra effort in the wider setting of Carnot groups, and we will therefore operate in this framework, which also allows for some simplifications in the notation. The presentation of Carnot groups will be only minimal, and we refer to [Reference Folland and Stein43, Reference Bonfiglioli, Lanconelli and Uguzzoni20, Reference Gromov58, Reference Le Donne66, Reference Serra Cassano88] for a more comprehensive treatment. The reader looking for a thorough account on intrinsic Lipschitz graphs might instead consult [Reference Franchi and Serapioni53].

2.1 Carnot groups: algebraic and metric preliminaries

A Carnot (or stratified) group is a connected, simply connected and nilpotent Lie group whose Lie algebra ${\mathfrak g}$ is stratified; that is, it possesses a decomposition ${\mathfrak g}={\mathfrak g}_{1}\oplus \dots \oplus {\mathfrak g}_{s}$ such that

$$ \begin{align*} \forall\ j=1,\dots,s-1\quad {\mathfrak g}_{j+1}=[{\mathfrak g}_{j},{\mathfrak g}_{1}],\qquad{\mathfrak g}_{s}\neq\{0\}\qquad\text{and}\qquad [{\mathfrak g}_{s},{\mathfrak g}]=\{0\}. \end{align*} $$

We refer to the integer s as the step of ${\mathbb G}$ and to $m:=\,$ dim ${\mathfrak g}_{1}$ as its rank; we also denote by d the topological dimension of ${\mathbb G}$ . The group identity is denoted by $0$ and, as is customary, we identify ${\mathfrak g}$ , $T_{0}{\mathbb G}$ and the algebra of left-invariant vector fields on ${\mathbb G}$ . The elements of ${\mathfrak g}_{1}$ are referred to as horizontal.

The exponential map $\exp :{\mathfrak g}\to {\mathbb G}$ is a diffeomorphism and, given a basis $X_{1},\dots ,X_{d}$ of ${\mathfrak g}$ , we will often identify ${\mathbb G}$ with ${\mathbb R}^{d}$ by means of exponential coordinates:

$$ \begin{align*} {\mathbb R}^{d}\ni x=(x_{1},\dots,x_{d})\longleftrightarrow \exp\left( x_{1}X_{1}+\dots+x_{d}X_{d}\right)\in{\mathbb G}. \end{align*} $$

We will also assume that the basis is adapted to the stratification; that is, that

$$ \begin{align*} & X_{1},\dots,X_{m} \text{ is a basis of }{\mathfrak g}_{1}\text{ and}\\ & \forall\ j=2,\dots,s,\ X_{\dim ({\mathfrak g}_{1}\oplus\cdots\oplus{\mathfrak g}_{j-1})+1},\dots, X_{\dim ({\mathfrak g}_{1}\oplus\cdots\oplus{\mathfrak g}_{j})}\text{ is a basis of }{\mathfrak g}_{j}\text{.} \end{align*} $$

In these coordinates, one has

(2.1) $$ \begin{align} X_{i}(x)=\partial_{x_{i}}+\sum_{j=m+1}^{d} P_{i,j}(x)\partial_{x_{j}}\quad\text{for every }i=1,\dots,m \end{align} $$

for suitable polynomial functions $P_{i,j}$ . A one-parameter family $\{\delta _{\lambda }\}_{\lambda>0}$ of dilations $\delta _{\lambda }:{\mathfrak g}\to {\mathfrak g}$ is defined by (linearly extending)

$$ \begin{align*} \delta_{\lambda}(X):=\lambda^{j} X\text{ for any }X\in{\mathfrak g}_{j}\text{;} \end{align*} $$

notice that dilations are Lie algebra homomorphisms and $\delta _{\lambda \mu }=\delta _{\lambda }\circ \delta _{\mu }$ . By composition with $\exp $ , one can then define a one-parameter family, for which we use the same symbol, of group isomorphisms $\delta _{\lambda }:{\mathbb G}\to {\mathbb G}$ .

We fix a left-invariant homogeneous distance d on ${\mathbb G}$ , so that

$$ \begin{align*} d(xy,xz)=d(y,z)\quad\text{and}\quad d(\delta_{\lambda} x,\delta_{\lambda} y)=\lambda d(x,y)\qquad\text{for all }x,y,z\in{\mathbb G},\lambda>0. \end{align*} $$

We use d to denote both the distance on ${\mathbb G}$ and its topological dimension, but no confusion will ever arise. We denote by $B(x,r)$ the open ball of center $x\in {\mathbb G}$ and radius $r>0$ ; it will also be convenient to denote by $\|\cdot \|_{{\mathbb G}}$ the homogeneous norm defined for $x\in {\mathbb G}$ by $\|x\|_{{\mathbb G}}:=d(0,x)$ . Recall that ${\mathscr L}^{d}$ is a Haar measure on ${\mathbb G}\equiv {\mathbb R}^{d}$ and that the homogeneous dimension of ${\mathbb G}$ is the integer $Q:=\sum _{j=1}^{s} j\dim {\mathfrak g}_{j}$ . One has

$$ \begin{align*} {\mathscr L}^{d} (B(x,r)) = r^{Q} {\mathscr L}^{d} (B(0,1))\qquad\text{for all }x\in{\mathbb G},r>0. \end{align*} $$

The number Q is always greater than d (apart from the Euclidean case $s=1$ ) and it coincides with the Hausdorff dimension of ${\mathbb G}$ . Since the Hausdorff Q-dimensional measure is also a Haar measure, it coincides with ${\mathscr L}^{d}$ up to a constant.

Given a measurable function $f:{\mathbb G}\to {\mathbb R}$ , we denote by $\nabla _{{\mathbb G}} f=(X_{1}f,\dots ,X_{m}f)$ its horizontal derivatives in the sense of distributions. It is well-known that, if f is Lipschitz continuous, then it is Pansu differentiable almost everywhere [Reference Pansu81] and, in particular, the pointwise horizontal gradient $\nabla _{{\mathbb G}} f$ exists almost everywhere on ${\mathbb G}$ . Moreover (see, e.g., [Reference Franchi, Serapioni and Serra Cassano47, Reference Garofalo and Nhieu56]), we have

(2.2) $$ \begin{align} \text{if }f:{\mathbb G}\to{\mathbb R}\text{ is continuous, then }\textit{f}\text{ is Lipschitz if and only if }\nabla_{{\mathbb G}} f\in L^{\infty}({\mathbb G}) \end{align} $$

where Lipschitz continuity, of course, is meant with respect to the homogeneous distance d on ${\mathbb G}$ . It is worth mentioning that the Lipschitz constant of f is bounded by $\|\nabla _{{\mathbb G}} f\|_{L^{\infty }({\mathbb G})}$ , apart from multiplicative constants that depend only on the distance d, both from below and from above.

We will need the following result later, proved in [Reference Vittone93, Lemma 2.2], where we denote by $\overrightarrow {\exp }(X)(x)$ the point reached in unit time by the integral curve of a vector field X starting at a point x.

Lemma 2.1. Let $f:{\mathbb R}^{d}\to {\mathbb R}$ be a continuous function and let Y be a smooth vector field in ${\mathbb R}^{d}$ . Assume that $Yf\geq \delta $ holds, in the sense of distributions, on an open set $U\subset {\mathbb R}^{d}$ and for a suitable $\delta \in {\mathbb R}$ . If $x\in U$ and $T>0$ are such that $\overrightarrow {\exp }(hY)(x)\in U$ for every $h\in [0,T)$ , then

$$ \begin{align*} f(\overrightarrow{\exp}(tY)(x))\geq f(x)+ \delta t\quad \text{ for every }t\in[0,T). \end{align*} $$

2.2 Intrinsic Lipschitz graphs

Following [Reference Franchi and Serapioni53], we fix a splitting ${\mathbb G}={\mathbb W}{\mathbb V}$ in terms of a couple ${\mathbb W},{\mathbb V}$ of homogeneous (i.e., invariant under dilations) and complementary (i.e., ${\mathbb W}\cap {\mathbb V}=\{0\}$ and ${\mathbb G}={\mathbb W}{\mathbb V}$ ) Lie subgroups of ${\mathbb G}$ . In exponential coordinates, ${\mathbb W},{\mathbb V}$ are linear subspaces of ${\mathbb G}\equiv {\mathbb R}^{d}$ . Clearly, the splitting induces for every $x\in {\mathbb G}$ a unique decomposition $x=x_{{\mathbb W}} x_{{\mathbb V}}$ such that $x_{{\mathbb W}}\in {\mathbb W}$ and $x_{{\mathbb V}}\in {\mathbb V}$ ; we will sometimes refer to the maps $x\mapsto x_{{\mathbb W}}$ and $x\mapsto x_{{\mathbb V}}$ as the projections of ${\mathbb G}$ on ${\mathbb W}$ and on ${\mathbb V}$ , respectively.

Given $A\subset {\mathbb W}$ and a map $\phi :A\to {\mathbb V}$ , the intrinsic graph $\mathrm {gr}_{\phi }$ of $\phi $ is the set

$$ \begin{align*} \mathrm{gr}_{\phi}:=\{w\phi(w):w\in A\}\subset{\mathbb G}. \end{align*} $$

The notion of intrinsic Lipschitz continuity for maps $\phi $ from ${\mathbb W}$ to ${\mathbb V}$ was introduced by B. Franchi, R. Serapioni and F. Serra Cassano [Reference Franchi, Serapioni and Serra Cassano50] in terms of a cone property for $\mathrm {gr}_{\phi }$ . The intrinsic cone ${\mathscr C}_{\alpha }$ of aperture $\alpha> 0$ and axis ${\mathbb V}$ is

$$ \begin{align*} {\mathscr C}_{\alpha}:=\{x\in{\mathbb G}:\|x_{{\mathbb W}}\|_{{\mathbb G}}\leq\alpha\|x_{{\mathbb V}}\|_{{\mathbb G}} \}\,. \end{align*} $$

Observe that ${\mathscr C}_{\alpha }$ is homogeneous (invariant under dilations) and that ${\mathbb V}\subset {\mathscr C}_{\alpha }$ . For $x\in {\mathbb G}$ we also introduce the cone ${\mathscr C}_{\alpha }(x):=x{\mathscr C}_{\alpha }$ with vertex x.

Definition 2.2. Let $A\subset {\mathbb W}$ ; we say that $\phi :A\to {\mathbb V}$ is intrinsic Lipschitz if there exists $\alpha>0$ such that

(2.3) $$ \begin{align} \forall\:x\in\mathrm{gr}_{\phi}\qquad \mathrm{gr}_{\phi}\cap {\mathscr C}_{\alpha}(x)=\{x\}. \end{align} $$

The intrinsic Lipschitz constant of $\phi $ is $\inf \{\frac 1\alpha :\alpha>0\text { and~(2.3) holds}\}$ .

Since all homogeneous distances on ${\mathbb G}$ are equivalent, Definition 2.2 is clearly independent from the fixed distance d on the group. It was proved in [Reference Franchi and Serapioni53, Theorem 3.9] that, if $\phi :{\mathbb W}\to {\mathbb V}$ is an entire intrinsic Lipschitz map, then the Hausdorff dimension of $\mathrm {gr}_{\phi }$ is the same as the Hausdorff dimension of the domain ${\mathbb W}$ ; actually, the corresponding Hausdorff measure on $\mathrm {gr}_{\phi }$ is Ahlfors regular and then also locally finite on $\mathrm {gr}_{\phi }$ . This implies that for entire intrinsic Lipschitz graphs one has

(2.4) $$ \begin{align} {\mathscr L}^{d}(\mathrm{gr}_{\phi})=0, \end{align} $$

provided, of course, we are not in the trivial case ${\mathbb W}={\mathbb G}$ , ${\mathbb V}=\{0\}$ . As one can imagine, the equality (2.4) holds, however, for every intrinsic Lipschitz $\phi :A\subset {\mathbb W}\to {\mathbb V}$ provided ${\mathbb W}\neq {\mathbb G}$ : one way of proving this fact is by noticing that ${\mathscr L}^{d}({\mathscr C}_{\alpha }(x)\cap B(x,r))$ is a fixed fraction of ${\mathscr L}^{d}( B(x,r))$ ; hence, $\mathrm {gr}_{\phi }$ cannot have points of density 1 and (2.4) follows by Lebesgue’s differentiation theorem in doubling metric spaces (see, e.g., [Reference Heinonen59, Theorem 1.8]).

Remark 2.3. For the purpose of future references, we observe the following easy fact. Let $S\subset {\mathbb G}$ and $\alpha>0$ be fixed; if

$$ \begin{align*} \forall\:x\in\ S\qquad S\cap {\mathscr C}_{\alpha}(x)=\{x\}, \end{align*} $$

then $S=\mathrm {gr}_{\phi }$ for suitable $\phi :A\to {\mathbb V}$ (which is clearly intrinsic Lipschitz) and $A\subset {\mathbb W}$ . See, for example, [Reference Franchi and Serapioni53, §2.2.3].

2.3 A level set definition of co-horizontal intrinsic Lipschitz graphs

From now on we assume that the splitting ${\mathbb W}{\mathbb V}$ of ${\mathbb G}$ is fixed in such a way that ${\mathbb V}$ is not trivial ( ${\mathbb V}\neq \{0\}$ ) and it is horizontal; that is, ${\mathbb V}\subset \exp ({\mathfrak g}_{1})$ . Of course, this poses some algebraic restrictions: for instance, ${\mathbb V}$ is forced to be abelian. Moreover, it can be easily checked that free Carnot groups (of step at least  $2$ ) have no splitting such that ${\mathbb V}$ is horizontal and $\dim {\mathbb V}\geq 2$ . Nonetheless, the theory we are going to develop here is rich enough to include intrinsic Lipschitz graphs of codimension 1 in any Carnot group (in fact, every 1-dimensional horizontal subgroup ${\mathbb V}$ of a Carnot group provides a splitting ${\mathbb W}{\mathbb V}$ for some ${\mathbb W}$ ) and intrinsic Lipschitz graphs of codimension at most n in the Heisenberg group ${\mathbb H}^{n}$ , which are the main object of study of the present article.

With such assumptions on the splitting ${\mathbb W}{\mathbb V}$ , intrinsic Lipschitz graphs $\Phi :A\subset {\mathbb W}\to {\mathbb V}$ will be called co-horizontal (see [Reference Antonelli, Di Donato, Don and Le Donne6]). We denote by k the topological dimension of ${\mathbb V}$ and we assume without loss of generality that the adapted basis $X_{1},\dots ,X_{d}$ of ${\mathfrak g}$ has been fixed in such a way that

$$ \begin{align*} {\mathbb V}=\exp(\operatorname*{\mathrm{span}}\{X_{1},\dots,X_{k}\}). \end{align*} $$

We consequently identify ${\mathbb V}$ with ${\mathbb R}^{k}$ through the map

(2.5) $$ \begin{align} {\mathbb R}^{k}\ni (v_{1},\dots,v_{k})\longleftrightarrow \exp(v_{1}X_{1}+\dots+v_{k}X_{k})\in{\mathbb V}, \end{align} $$

and we accordingly write $ v=(v_{1},\dots ,v_{k})\in {\mathbb V}$ . The map in (2.5) turns out to be a group isomorphism as well as a bi-Lipschitz map between $({\mathbb R}^{k},|\cdot |)$ and $({\mathbb V},d)$ : this proves that the Hausdorff dimension of ${\mathbb V}$ equals the topological dimension k. We observe that, since the flow of a left-invariant vector field corresponds to right multiplication, we have

$$ \begin{align*} xv=\overrightarrow{\exp} (v_{1}X_{1}+\dots+v_{k}X_{k})(x). \end{align*} $$

In particular, the projections on the factors ${\mathbb W},{\mathbb V}$ can be written as

$$ \begin{align*} x_{{\mathbb V}}=\exp(x_{1}X_{1}+\dots+x_{k}X_{k}),\qquad x_{{\mathbb W}} =x\,x_{{\mathbb V}}^{-1}=\overrightarrow{\exp}(-(x_{1}X_{1}+\dots+x_{k}X_{k}))(x) \end{align*} $$

and are therefore smooth maps.

Our first goal is to provide the equivalent characterisation of co-horizontal intrinsic Lipschitz graphs stated in Theorem 1.4. However, we need some preparatory lemmata as well as some extra convention about notation. First, we introduce the homogeneous (pseudo)-norm

$$ \begin{align*} \|x\|_{*}:=\left(\sum_{j=1}^{s} \sum_{i:X_{i}\in{\mathfrak g}_{j}}|x_{i}|^{\frac{2\;\!s!}{j}}\right)^{\frac1{2\;\!s!}},\qquad x\in{\mathbb G}, \end{align*} $$

which is equivalent to $\|\cdot \|_{{\mathbb G}}$ in the sense that there exists $C_{*}\geq 1$ such that

(2.6) $$ \begin{align} \|x\|_{{\mathbb G}}/C_{*}\leq \|x\|_{*}\leq C_{*} \|x\|_{{\mathbb G}}\qquad\forall\ x\in{\mathbb G}. \end{align} $$

Observe that $x\mapsto \|x\|_{*}$ is of class $C^{\infty }$ in ${\mathbb G}\setminus \{0\}$ . Second, given $i\in \{1,\dots ,k\}$ , $\beta>0$ and $\varepsilon>0$ , we introduce the homogeneous cone

$$ \begin{align*} \begin{aligned} {\mathscr C}_{i,\beta,\varepsilon} :=& \left\{wv:w\in{\mathbb W},\ v\in{\mathbb V},\ |v_{i}|+\varepsilon\sum_{j\in\{1,\dots,k\}\setminus\{i\}}|v_{j}|\geq \beta\|w\|_{*}\right\}\\ = & \left\{x\in{\mathbb G}:|x_{i}|+\varepsilon\sum_{j\in\{1,\dots,k\}\setminus\{i\}}|x_{j}|\geq \beta\|x_{{\mathbb W}}\|_{*}\right\}, \end{aligned} \end{align*} $$

where we used the fact that $x_{{\mathbb V}}=(x_{1},\dots ,x_{k})$ . Third, if $t\in {\mathbb R}$ and $f:D\to {\mathbb R}$ is a real-valued function defined on some set D, we denote by $\{f\geq t\}$ the set $\{x\in D:f(x)\geq t\}$ . Similar conventions are understood when writing $\{f>t\},\ \{f<t\},\ \{f=t\},\ \{t_{1}<f<t_{2}\}$ , etc.

Lemma 2.4. For every $i\in \{1,\dots ,k\}$ , $\beta>0$ and $\varepsilon \in (0,1)$ there exists a 1-homogeneous Lipschitz function $f_{i,\beta ,\varepsilon }:{\mathbb G}\to {\mathbb R}$ such that

(2.7) $$ \begin{align} f_{i,\beta,\varepsilon}(0)=0 \hphantom{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\end{align} $$
(2.8) $$ \begin{align} 1\leq X_{i}f_{i,\beta,\varepsilon}\leq 3\quad{\mathscr L}^{d}\text{-a.e. on }{\mathbb G} \hphantom{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\end{align} $$
(2.9) $$ \begin{align} \varepsilon\leq X_{\ell} f_{i,\beta,\varepsilon}\leq 3\varepsilon\quad{\mathscr L}^{d}\text{-a.e. on }{\mathbb G}\quad \forall\ \ell\in\{1,\dots,k\}\setminus\{i\} \end{align} $$
(2.10) $$ \begin{align} \{f_{i,\beta,\varepsilon}\geq 0\} \subset {\mathscr C}_{i,\beta,\varepsilon}.\hphantom{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \end{align} $$

Moreover, if $0<\beta \leq \bar \beta $ , then the Lipschitz constant of $f_{i,\beta ,\varepsilon }$ can be controlled in terms of $\varepsilon $ and $\bar \beta $ only.

Proof. Without loss of generality, we can assume that $i=1$ . For $x\in {\mathbb G}$ , define

$$ \begin{align*} f(x):=\left\{ \begin{array}{l@{\quad}l} 2[x_{1}+\varepsilon(x_{2}+\dots+x_{k})-\beta\| x_{{\mathbb W}}\|_{*}]\quad & \text{if }|x_{1}+\varepsilon(x_{2}+\dots+x_{k})|\leq 2\beta\| x_{{\mathbb W}}\|_{*} \\ (x_{1}+\varepsilon(x_{2}+\dots+x_{k})) & \text{if }x_{1}+\varepsilon(x_{2}+\dots+x_{k})> 2\beta\| x_{{\mathbb W}}\|_{*}\\ 3[x_{1}+\varepsilon(x_{2}+\dots+x_{k})] & \text{if }x_{1}+\varepsilon(x_{2}+\dots+x_{k})<- 2\beta\| x_{{\mathbb W}}\|_{*}. \end{array} \right. \end{align*} $$

We prove that $f_{1,\beta ,\varepsilon }:=f$ satisfies all of the claimed statements. Property (2.7) and the homogeneity of f are immediate. Property (2.10) is equivalent to the implication

$$ \begin{align*} |x_{1}|+\varepsilon(|x_{2}|+\dots+|x_{k}|)< \beta\|x_{{\mathbb W}}\|_{*}\ \Longrightarrow\ f(x)<0, \end{align*} $$

which one can easily check. The function f is continuous on ${\mathbb G}$ and smooth on ${\mathbb G}\setminus D$ , where

$$ \begin{align*} D:=\{x\in{\mathbb G}:|x_{1}+\varepsilon(x_{2}+\dots+x_{k})| = 2\beta\| x_{{\mathbb W}}\|_{*} \}. \end{align*} $$

Since D is ${\mathscr L}^{d}$ -negligible, statements (2.8) and (2.9) follow if we prove that for every $\ell =2,\dots ,k$ ,

(2.11) $$ \begin{align} 1\leq X_{1}f\leq 3\quad\text{and}\quad \varepsilon\leq X_{\ell} f\leq 3\varepsilon\qquad\text{on }{\mathbb G}\setminus D. \end{align} $$

Using (2.1) one gets

(2.12) $$ \begin{align} \begin{array}{l@{\quad }l} \nabla_{{\mathbb G}} f(x)=(1,\varepsilon,\dots,\varepsilon,0,\dots,0) & \text{if }x_{1}+\varepsilon(x_{2}+\dots+x_{k})> 2\beta\| x_{{\mathbb W}}\|_{*}\\ \nabla_{{\mathbb G}} f(x)=(3,3\varepsilon,\dots,3\varepsilon,0,\dots,0)\quad & \text{if }x_{1}+\varepsilon(x_{2}+\dots+x_{k})<- 2\beta\| x_{{\mathbb W}}\|_{*}\,. \end{array} \end{align} $$

We now notice that, for any $x\in {\mathbb G}$ , the map $y\mapsto y_{{\mathbb W}}$ is constant on the coset $x{\mathbb V}$ , which is a smooth submanifold tangent to $X_{1},\dots ,X_{k}$ . This implies that

$$ \begin{align*} (X_{1}f,\dots,X_{k}f)(x)=(2,2\varepsilon,\dots,2\varepsilon)\qquad\text{if }|x_{1}+\varepsilon(x_{2}+\dots+x_{k})|<2\beta\| x_{{\mathbb W}}\|_{*}, \end{align*} $$

which, together with (2.12), implies (2.11).

We have only to check that f is Lipschitz continuous on ${\mathbb G}$ and that a bound on the Lipschitz constant can be given in terms of $\varepsilon $ and $\bar \beta $ . Taking into account (2.12) and the continuity of f on ${\mathbb G}$ , by (2.2) it is enough to prove the function $g:{\mathbb G}\to {\mathbb R}$ defined by

$$ \begin{align*} g(x):=2[x_{1}+\varepsilon(x_{2}+\dots+x_{k})-\beta\| x_{{\mathbb W}}\|_{*}] \end{align*} $$


(2.13) $$ \begin{align} |\nabla_{{\mathbb G}} g|\leq C\quad\text{on }\{x\in{\mathbb G}:|x_{1}+\varepsilon(x_{2}+\dots+x_{k})|<2\beta\| x_{{\mathbb W}}\|_{*}\} \end{align} $$

for some positive C. Since $x\mapsto x_{{\mathbb W}}$ is smooth on ${\mathbb G}$ and $\|\cdot \|_{*}$ is smooth on ${\mathbb G}\setminus \{0\}$ , we get that g is smooth on ${\mathbb G}\setminus {\mathbb V}$ . Moreover, g is 1-homogeneous; thus, $\nabla _{{\mathbb G}} g$ is 0-homogeneous (i.e., invariant under dilations) and continuous on ${\mathbb G}\setminus {\mathbb V}$ . Inequality (2.13) will then follow if we prove that

$$ \begin{align*} |\nabla_{{\mathbb G}} g|\leq C\quad\text{on } \partial B(0,1)\cap\{x\in{\mathbb G}:|x_{1}+\varepsilon(x_{2}+\dots+x_{k})|\leq\beta\| x_{{\mathbb W}}\|_{*}\}; \end{align*} $$

in turn, this inequality and the bound (in terms of $\varepsilon ,\bar \beta $ ) on the Lipschitz constant of f follow by proving that

(2.14) $$ \begin{align} |\nabla_{{\mathbb G}} g|\leq C\quad\text{on } \partial B(0,1)\cap\{x\in{\mathbb G}:|x_{1}+\varepsilon(x_{2}+\dots+x_{k})|\leq\bar\beta\| x_{{\mathbb W}}\|_{*}\}. \end{align} $$

The set ${\mathbb V}$ is closed, while $\partial B(0,1)\cap \{x\in {\mathbb G}:|x_{1}+\varepsilon (x_{2}+\dots +x_{k})|\leq \bar \beta \| x_{{\mathbb W}}\|_{*}\}$ is compact; since they are disjoint, they have positive distance and the continuity of $\nabla _{{\mathbb G}} g$ on ${\mathbb G}\setminus {\mathbb V}$ ensures that

$$ \begin{align*} \sup \big\{|\nabla_{{\mathbb G}} g(x)|:x\in\partial B(0,1)\text{ and }|x_{1}+\varepsilon(x_{2}+\dots+x_{k})|\leq \bar\beta\| x_{{\mathbb W}}\|_{*}\big\} < +\infty, \end{align*} $$

which is (2.14) and allows us to conclude.

Lemma 2.5. Let $A\subset {\mathbb W}$ be nonempty and let $\phi :A\to {\mathbb V}$ be intrinsic Lipschitz. Then for every $\varepsilon \in (0,1)$ and $i\in \{1,\dots ,k\}$ there exists a Lipschitz function $f_{i,\varepsilon }:{\mathbb G}\to {\mathbb R}$ such that

(2.15) $$ \begin{align} & \mathrm{gr}_{\phi}\subset\{f_{i,\varepsilon}=0\} \hphantom{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\end{align} $$
(2.16) $$ \begin{align} & 1\leq X_{i}f_{i,\varepsilon}\leq 3 \qquad{\mathscr L}^{d}\text{-a.e. on }{\mathbb G} \hphantom{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\end{align} $$
(2.17) $$ \begin{align} & \varepsilon \leq X_{\ell} f_{i,\varepsilon}\leq 3\varepsilon\qquad{\mathscr L}^{d}\text{-a.e. on }{\mathbb G}\quad \forall\ell\in\{1,\dots,k\}\setminus\{i\}. \end{align} $$

Moreover, if the intrinsic Lipschitz constant of $\phi $ is not greater than $\bar \alpha>0$ , then the Lipschitz constant of $f_{i,\varepsilon }$ can be bounded in terms of $\varepsilon $ and $\bar \alpha $ only.

Proof. Assume that the intrinsic Lipschitz constant of $\phi $ is not greater than some $\bar \Lambda>0$ and define $\alpha :=(2\Lambda )^{-1}$ ; then (2.3) holds for such $\alpha $ . Recalling that the constant $C_{*}>0$ was introduced in (2.6), we set $\beta :=k\:\!C_{*}^{2}/\alpha $ . Taking into account the inequalities

$$ \begin{align*} & |x_{i}|+\varepsilon\!\!\!\sum_{j\in\{1,\dots,k\}\setminus\{i\}}\!\!\!\!|x_{j}|\ \leq |x_{1}|+\dots+|x_{k}|\leq k\|x_{{\mathbb V}}\|_{*}\leq kC_{*}\|x_{{\mathbb V}}\|_{{\mathbb G}},\\ & \beta\|x_{{\mathbb W}}\|_{*} \geq \beta\|x_{{\mathbb W}}\|_{{\mathbb G}}/{C_{*}} \end{align*} $$

we obtain the inclusion ${\mathscr C}_{i,\beta ,\varepsilon }\subset {\mathscr C}_{\alpha }$ . For $y\in {\mathbb G}$ , set $f_{y}(x):=f_{i,\beta ,\varepsilon }(y^{-1}x)$ , where $f_{i,\beta ,\varepsilon }$ is the function provided by Lemma 2.4, and define

$$ \begin{align*} f(x):=\sup_{y\in \mathrm{gr}_{\phi}} f_{y}(x). \end{align*} $$

We prove that $f_{i,\varepsilon }:=f$ satisfies the claimed statement.

Let $x\in \mathrm {gr}_{\phi }$ ; then $f(x)\geq f_{x}(x)=0$ , while for every $y\in \mathrm {gr}_{\phi }\setminus \{x\}$ one has $f_{y}(x)<0$ because of (2.10) and

$$ \begin{align*} \mathrm{gr}_{\phi}\cap \{f_{y}\geq0\}\ =\ \mathrm{gr}_{\phi}\cap y \{f_{i,\beta,\varepsilon}\geq0\} \subset\ \mathrm{gr}_{\phi}\cap y {\mathscr C}_{i,\beta,\varepsilon}\ \subset\ \mathrm{gr}_{\phi}\cap y{\mathscr C}_{\alpha}\ =\ \{y\}. \end{align*} $$

This proves that $f(x)=0$ , which is (2.15).

The functions $f_{y}$ are uniformly Lipschitz continuous; hence, f shares the same Lipschitz continuity. Let $x\in {\mathbb G}$ be fixed; then for every $\eta>0$ there exists $y\in \mathrm {gr}_{\phi }$ such that

$$ \begin{align*} f_{y}(x)\geq f(x)-\eta. \end{align*} $$

Since $X_{i} f_{y}\geq 1$ , by Lemma 2.1 we have for every $t\geq 0$

$$ \begin{align*} f(\overrightarrow{\exp}(tX_{i})(x))\geq f_{y}(\overrightarrow{\exp}(tX_{i})(x))\geq f_{y}(x) +t\geq f(x)+t -\eta. \end{align*} $$

By the arbitrariness of $\eta $ , one obtains

$$ \begin{align*} f(\overrightarrow{\exp}(tX_{i})(x))\geq f(x)+t\quad\text{for every }t\geq 0; \end{align*} $$

that is, $X_{i}f\geq 1$ a.e. on ${\mathbb G}$ . A similar argument, using $g:=-f$ and the inequality $X_{i}g\geq -3$ , shows that $f(\overrightarrow {\exp }(tX_{i})(x))\leq f(x)+3t$ for every $t\geq 0$ ; that is, that $X_{i}f\leq 3$ a.e. on ${\mathbb G}$ . This proves (2.16).

The proof of (2.17) is completely analogous and we omit it.

The following lemma is most likely well-known; however, we provide a proof for the sake of completeness.

Lemma 2.6. Let $f:{\mathbb R}^{k}\to {\mathbb R}^{k}$ be a Lipschitz map such that there exists $\delta>0$ for which

(2.18) $$ \begin{align} \langle f(x+v)-f(x),v\rangle \geq \delta|v|^{2}\qquad\text{for every }x,v\in{\mathbb R}^{k}. \end{align} $$

Then there exists a unique $\bar {x}\in {\mathbb R}^{k}$ such that $f(\bar {x})=0$ .

Proof. We reason by induction on k and leave the case $k=1$ as an exercise to the reader. We assume that the lemma holds for some $k\geq 1$ and we prove it for $k+1$ .

By the 1-dimensional case of the lemma, for every $x\in {\mathbb R}^{k}$ there exists a unique $g(x)\in {\mathbb R}$ such that $f_{k+1}(x,g(x))=0$ ; we claim that g is Lipschitz continuous. Letting L denote the Lipschitz constant of f, we indeed have for every $x,y\in {\mathbb R}^{k}$

$$ \begin{align*} f_{k+1}(y,g(x)+\tfrac L\delta|y-x|) \stackrel{(2.18)}{\geq} & f_{k+1}(y,g(x)) + L|y-x|\\ \stackrel{\phantom{(2.18)}}{\geq} & f_{k+1}(x,g(x)) - L|y-x| + L|y-x|\ =\ 0 \end{align*} $$


$$ \begin{align*} f_{k+1}(y,g(x)-\tfrac L\delta|y-x|) \stackrel{(2.18)}{\leq} & f_{k+1}(y,g(x)) - L|y-x|\\ \stackrel{\phantom{(2.18)}}{\leq} & f_{k+1}(x,g(x)) + L|y-x| - L|y-x|\ =\ 0. \end{align*} $$

The last two displayed formulae imply that

$$ \begin{align*} g(x)-\tfrac L\delta|y-x| \leq g(y) \leq g(x)+\tfrac L\delta|y-x|, \end{align*} $$

and the Lipschitz continuity of g follows. In particular, the function $h:{\mathbb R}^{k}\to {\mathbb R}^{k}$ defined by $h(z):=(f_{1},\dots ,f_{k})(z,g(z))$ is Lipschitz continuous; since

$$ \begin{align*} \langle h(z+v)-h(z),v\rangle \geq \delta|v|^{2}\qquad\text{for every }z,v\in{\mathbb R}^{k}, \end{align*} $$

by inductive assumption there is a unique $\bar z\in {\mathbb R}^{k}$ such that $h(\bar z)=0$ . It follows that $\bar {x}:=(\bar z,g(\bar z))$ is the unique zero of f, which concludes the proof.

Before passing to the the main proof of this section, we recall once more that ${\mathbb V}$ is identified with ${\mathbb R}^{k}$ by

$$ \begin{align*} {\mathbb R}^{k}\ni(v_{1},\dots,v_{k})\longleftrightarrow \exp(v_{1}X_{1}+\dots+v_{k}X_{k})\in {\mathbb V}. \end{align*} $$

This identification is understood, in particular, when considering scalar products between elements of ${\mathbb R}^{k}$ and ${\mathbb V}$ as in (1.2).

Remark 2.7. It is easily seen that, for a given Lipschitz map $f:{\mathbb G}\to {\mathbb R}^{k}$ , statement (1.2) is equivalent to the uniform ellipticity (a.k.a. coercivity) of the matrix col $[X_{1}f|\dots |X_{k} f]$ ; that is, to the fact that

(2.19) $$ \begin{align} \text{col}[X_{1}f|\dots|X_{k} f](x)\geq \delta\: I\qquad\text{for }{\mathscr L}^{d}\text{-a.e. }x\in{\mathbb G} \end{align} $$

in the sense of bilinear forms, where I denotes the $k\times k$ identity matrix. Observe that such a matrix is defined a.e. on ${\mathbb G}$ by Pansu’s theorem [Reference Pansu81, Théorème 2].

Proof of Theorem 1.4.

Step 1. We prove the implication (a) $\Rightarrow $ (b). Consider the map

$$ \begin{align*} f:=(f_{1,\varepsilon},\dots,f_{k,\varepsilon}):{\mathbb G}\to{\mathbb R}^{k}, \end{align*} $$

where $\varepsilon \in (0,1)$ will be determined later and the functions $f_{i,\varepsilon }$ are provided by Lemma 2.5. The inclusion (1.1) follows from (2.15). In order to prove (1.2), we first observe that for ${\mathscr L}^{d}$ -a.e. $x\in {\mathbb G}$ and ${\mathscr L}^{k}$ -a.e. $v\in {\mathbb R}^{k}\equiv {\mathbb V}$ one has

$$ \begin{align*} \langle f(xv)-f(x),v\rangle =& \int_{0}^{1} \left\langle \sum_{j=1}^{k}v_{j}X_{j}f(x\delta_{t}v),v\right\rangle\:dt, \end{align*} $$

where we used the fact that $x\delta _{t}v=\overrightarrow {\exp }(t(v_{1}X_{1}+\dots +v_{k}X_{k}))(x)$ . Therefore,

$$ \begin{align*} \langle f(xv)-f(x),v\rangle = & \int_{0}^{1} \sum_{i,j=1}^{k}v_{i}v_{j}X_{j}f_{i}(x\delta_{t}v) \:dt\\ = & \int_{0}^{1} \sum_{i=1}^{k}v_{i}^{2}X_{i}f_{i}(x\delta_{t}v) \:dt + \int_{0}^{1} \sum_{\substack{i,j=1,\dots, k\\i\neq j}}v_{i}v_{j}X_{j}f_{i}(x\delta_{t}v) \:dt\\ \geq & (1-3(k^{2}-k)\varepsilon) |v|^{2}, \end{align*} $$

where in the last inequality we used (2.16) and (2.17). If $k=1$ , this inequality is (1.2) with $\delta =1$ ; if $k\geq 2$ , (1.2) follows with $\delta =1/2$ provided we choose $ \varepsilon =(6(k^{2}-k))^{-1}$ . This proves the implication (a) $\Rightarrow $ (b).

Step 2. We now prove the converse implication (b) $\Rightarrow $ (a); it is enough to prove that $Z_{f}:=\{f=0\}$ is the intrinsic graph of some intrinsic Lipschitz function $\phi :{\mathbb W}\to {\mathbb V}$ . For every $w\in {\mathbb W}$ , define $f_{w}:{\mathbb V}\equiv {\mathbb R}^{k}\to {\mathbb R}^{k}$ as $f_{w}(v):=f(wv)$ . By Lemma 2.6 there is a unique $\bar v=\bar v(w)$ such that $f_{w}(\bar v)=0$ ; we define $\phi :{\mathbb W}\to {\mathbb V}$ by $\phi (w):=\bar v$ . For $\lambda \in (0,1)$ , which will be fixed later, we introduce the homogeneous cone

$$ \begin{align*} D_{\lambda}:=\bigcup_{v\in{\mathbb V}} \overline{B(v,\lambda \|v\|_{{\mathbb G}})}=\bigcup_{v\in{\mathbb V}} v\overline{B(0,\lambda \|v\|_{{\mathbb G}})}. \end{align*} $$

By a simple topological argument (see, e.g., [Reference Don, Massaccesi and Vittone39, Remark A.2]) there exists $\alpha =\alpha (\lambda )>0$ such that ${\mathscr C}_{\alpha }\subset D_{\lambda }$ ; in order to prove that $\phi $ is intrinsic Lipschitz, it is sufficient to show that

(2.20) $$ \begin{align} Z_{f}\cap xD_{\lambda} =\{x\}\qquad\forall\; x\in Z_{f}. \end{align} $$

To this aim, for every $x\in Z_{f}$ and every $y\in xD_{\lambda }\setminus \{x\}$ , one has by definition

$$ \begin{align*} y=xvz\qquad \text{for some }v\in{\mathbb V}\setminus\{0\}\text{ and }z\in{\mathbb G}\text{ such that }d(0,z)\leq\lambda d(0,v). \end{align*} $$

Denoting by L the Lipschitz constant of f we obtain

$$ \begin{align*} \langle f(y),v\rangle = & \langle f(xvz)-f(xv),v\rangle + \langle f(xv)-f(x),v\rangle\\ \geq & - L\,d(0,z)|v| + \delta|v|^{2} \\ \geq & -L\lambda\,d(0,v)|v|+ \delta|v|^{2}\\ \geq & -L\lambda\, C_{*}\|v\|_{*}|v|+ \delta|v|^{2}\\ \geq & (\delta-L\lambda C_{*}c_{k})|v|^{2} \end{align*} $$

for some positive constant $c_{k}$ depending on k only. It follows that, provided $\lambda $ is chosen small enough, one has $\langle f(y),v\rangle>0$ ; hence, $f(y)\neq 0$ and $y\notin Z_{f}$ . This proves (2.20) and concludes the proof of the theorem.

Remark 2.8. In Step 2 of the previous proof we showed that, if f is as in Theorem 1.4 (b), then the level set $\{f=0\}$ is an entire intrinsic Lipschitz graph; that is, it is the intrinsic graph of a ${\mathbb V}$ -valued map $\phi $ defined on the whole ${\mathbb W}$ .

Remark 2.9. It is worth pointing out that, in the implication (b) $\Rightarrow $ (a), the aperture $\alpha $ depends, apart from geometric quantities, only on the Lipschitz and coercivity constants $L,\delta $ of f. More precisely, if f is as in Theorem 1.4 (b), the Lipschitz constant L of f is not greater than some $\bar L>0$ and the coercivity constant $\delta $ is not smaller than some $\bar \delta>0$ , then the aperture $\alpha $ (and hence the intrinsic Lipschitz constant of $\phi $ ) can be controlled in terms of $\bar L$ and $\bar \delta $ only.

A similar remark applies at the level of the implication (a) $\Rightarrow $ (b); in fact, if $\phi $ is as in Theorem 1.4 (a) and the intrinsic Lipschitz constant of $\phi $ is not greater than some positive