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.

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).

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 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].

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,

(1.3)

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 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.