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Blow-up of regular submanifolds in Heisenberg groups and applications

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.


Introduction
In recent years, several efforts have been devoted to the project of developing Analysis and Geometry in stratified groups and more general Carnot-Carathéodory spaces with several monographs and surveys on this subject. Among them we mention [3], [6], [13], [16], [23], [28], but this list could be surely enlarged.
Our study fits into the recent project of developing Geometric Measure Theory in these spaces. Ambient of our investigations is the (2n+1)-dimensional Heisenberg group H n , which represents the simplest model of non-Abelian stratified group, [6], [27]. Aim of this paper is to present an intrinsic blow-up theorem for C 1 submanifolds in the geometry of the Heisenberg group along with its applications. The main feature of this procedure is the use of natural dilations of the group, namely, a one-parameter family of group homomorphisms that are homogeneous with respect to the distance of the group. Recall that dilations in H n are anisotropic, hence they differently act on different directions of the submanifold. The foremost directions are the so-called horizontal directions, that determine the "sub-Riemannian geometry" of the Heisenberg group: at any point x ∈ H n a 2n-dimensional subspace H x H n ⊂ T x H 2n+1 is given and the family of all horizontal spaces H x H n forms the so-called horizontal subbundle HH n . We will defer full definitions to Section 2.
The blow-up procedure consists in enlarging the submanifold Σ at some point x ∈ Σ by intrinsic dilations and taking the intersection of the magnified submanifold with a bounded set centered at x. We are interested in studying the case when T x Σ ⊂ H x H n , namely, x is a transverse point. The effect of rescaling the submanifold at a transverse point x can be obtained by considering the behavior of vol p (B x,r ∩ Σ)/r p+1 as r → 0 + , that heuristically is vol p (B x,r ∩ Σ) r p+1 = vol p (l x δ r (B 1 ∩ Σ x,r )) r p+1 = vol p (δ r (B 1 ∩ Σ x,r )) r p+1 ≈ α(x) vol p (B 1 ∩ Σ x,r ).
Here vol p denotes the p-dimensional Riemannian measure restricted to Σ, the left translation l x : H n −→ H n is given by l x (y) = x · y, the dilation of factor r > 0 is δ r : H n −→ H n , the dilated submanifold at x is Σ x,r = δ 1/r (l x −1 Σ) and B x,r is the open ball of center x and radius r with respect to a fixed homogeneous distance. The meaning of α(x) will be clear in the following theorem, that makes rigorous our previous consideration and represents our first main result.
where Ω is an open subset of H n and let x be a transverse point. Then the following limit holds A novel object appearing in this limit is the vertical tangent p-vector τ Σ,V (x), introduced in Definition 2.13. Its associated p-dimensional subspace of h n is a subalgebra whose image through the exponential map represents the blow-up limit of the rescaled submanifold Σ x,r as r → 0 + . The p-vector τ Σ,V (x) in higher codimension plays the same role that the well known horizontal normal ν H plays in codimension one (compare for instance with [19]). The metric factor θ(τ Σ,V (x)), introduced in [19], corresponds to the measure of the intersection of B 1 with the vertical subspace associated to the vertical tangent p-vector τ Σ,V (x). A first consequence of Theorem 1.1 is an explicit formula to compute the (p+1)-dimensional spherical Hausdorff measure of p-dimensional C 1 submanifolds in the Heisenberg group. In fact, thanks to S p+1 -negligibility of characteristic points proved in [22], Theorem 1.1 along with standard theorems on differentation of measures, immediately give the following result.
Theorem 1.2 Let ρ be a homogeneous distance with constant metric factor α > 0 and let S p+1 H n = α S p+1 ρ . Then we have Note that in codimension one, the integral formula (2)  Another consequence of Theorem 1.1 is the validity of an intrinsic coarea formula for vector-valued Lipschitz mappings defined on the Heisenberg group. By Sard theorem and the classical Whitney approximation theorem we can assume that a.e. level set is a submanifold of class C 1 , then we apply representation formula (2). The core of the proof stands in the key relation which surprisingly connects vertical tangent p-vector with horizontal jacobian J H f . The proof of (3) is given in Theorem 3.3. Thus, we can establish the following result.
where A ⊂ H n is a measurable subset and 1 ≤ k < 2n + 1. Let ρ be a homogeneous distance with constant metric factor α > 0. Then for every measurable function u : This coarea formula along with that of [21], which is a particular case, represent first examples of intrinsic coarea formulae for vector valued mappings defined on non-Abelian Carnot groups. It remains an interesting open question the extension of coarea formula to Lipschitz mappings with respect to a homogeneous distance. Only in the case of real-valued mappings this problem has been settled in [20]. This question is intimately related to a blow-up theorem of "intrinsicly regular" submanifolds. In this connection, we mention a recent work by Franchi, Serapioni and Serra Cassano [10], where a notion of intrinisic submanifold in H n has been introduced in arbitrary codimension. According the their terminology, a k-codimensional H-regular submanifold for algebraic reasons must satisfy 1 ≤ k ≤ n. With this restriction it might be highly irregular, even unrectifiable in the Euclidean sense, [14]. Nevertheless they show that an area-type formula for its (p+1)-dimensional spherical Hausdorff measure still holds. Here we wish to emphasize the difference in our approach, where we consider C 1 submanifolds, but with no restriction on their codimension. Let us summarize the contents of the present paper. Section 2 recalls some notions. Section 3 is devoted to the proof of Theorem 1.1. In Section 4 we show the validity of Theorem 1.2, along with its applications. Precisely, in Theorem 4.9 we show how a suitable rescaling of the spherical Hausdorff measure yields an intrinsic surface measure only depending on the sub-Riemannian metric, namely, the restriction of the Riemannian metric to the horizontal subbundle. In Proposition 4.5, we single out a privileged class of homogeneous distances having constant metric factor. We present several explicit computations of (p+1)-dimensional spherical Hausdorff measure in concrete examples. As another application of Theorem 1.2, we show a lower semicontintuity result for the spherical Hausdorff measure with respect to weak convergence of regular currents. Section 5 establishes an intrinsic coarea formula for vector-valued Riemannian Lipschitz mappings on the Heisenberg group.
The (2n+1)-dimensional Heisenberg group H n is a simply connected Lie group whose Lie algebra h n is equipped with a basis (X 1 , . . . , X 2n , Z) satisfying the bracket relations for every k = 1, . . . , n. We will identify the Lie algebra h n with the isomorphic Lie algebra of left invariant vector fields on H n , so that any X j also denotes a left invariant vector field of H n . In the terminology of Differential Geometry, the basis (X 1 , . . . , X 2n , Z) forms a moving frame in H n . We will say that (X 1 , . . . , X 2n , Z) is our standard frame. In particular, (X 1 , . . . , X 2n ) is a horizontal frame and it spans a smooth distribution of 2n-dimensional hyperplanes, called horizontal hyperplanes and denoted by H x H n for every x ∈ H n . The collection of all horizontal hyperplanes forms the so called horizontal subbundle, denoted by HH n . In the sequel, we will fix the unique left invariant Riemannian metric g such that the standard frame (X 1 , X 2 , . . . , X 2n , Z) forms an orthonormal basis at each point.
Definition 2.1 Every set of left invariant vector fields (Y 1 , . . . , Y 2n ) spanning the horizontal hyperplane at the unit element of H n will be called horizontal frame.
Recall that the exponential map exp : h n −→ H n is a diffeomorphism, then it is possible to introduce a system of coordinates in all of H n .
Definition 2.2 (Graded coordinates) Let (Y 1 , . . . , Y 2n ) be a horizontal frame and let W be a non horizontal left invariant vector field. The frame (Y 1 , . . . , Y 2n , W ) defines a coordinate chart F : R 2n+1 −→ H n given by Coordinates defined by (6) are called graded coordinates in the case W = Z and standard coordinates in the case the standard frame (X 1 , . . . , X 2n , Z) is used. In general we will say that the coordinates are associated to the frame (Y 1 , . . . , Y 2n , W ) We will assume throughout that a system of standard coordinate is fixed, if not stated otherwise.

Remark 2.3
Note that the horizontal frame (Y 1 , . . . , Y 2n ) of Definition 2.2 may not satisfy relations (5), where X i are replaced by Y i .
The standard frame with respect to standard coordinates reads as follows and the group operation is given by the following formula A natural family of dilations which respects the group operation (8) can be defined as follows for every r > 0. In fact, the map δ r : H n −→ H n defined above is a group homomorphism with respect to the operation (8).
In contrast with Analysis in Euclidean spaces, where the Euclidean distance is the most natural choice, in the Heisenberg group several distances have been introduced for different purposes. However, all of them are homogeneous in the following sense. If ρ : H n × H n −→ [0, +∞ + [ is a homogeneous distance, then 1. ρ is a continuous with respect to the topology of H n , 2. ρ(xy, xz) = ρ(y, z) for every x, y, z ∈ H n , 3. ρ(δ r y, δ r z) = r ρ(y, z) for every y, z ∈ H n and every r > 0.
To simplify notations we write ρ(x, 0) = ρ(x), where 0 denotes either the origin of R 2n+1 or the unit element of H n . The open ball of center x and radius r > 0 with respect to a homogeneous distance is denoted by B x,r . The Carnot-Carathéodory distance is an important example of homogeneous distance, [11]. However, all of our computations hold for a general homogeneous distance, therefore in the sequel ρ will denote a homogeneous distance, if not stated otherwise. Note that the Hausdorff dimension of H n with respect to any homogeneous distance is 2n + 2. Next, we recall the notion of Riemannian jacobian.
is understood with respect to the induced scalar products on Λ d (T x M ) and Λ k (T f (x) N ). We recall scalar products of p-vectors in (17).
To compute the Riemannian jacobian, we fix two orthonormal bases (X 1 , . . . , X d ) and (E 1 , . . . , E k ) of T x M and T f (x) N , respectively, and we represent df (x) with respect to these bases by the matrix Then the jacobian of the matrix ∇ X,E f (x) coincides with J g f (x). In the sequel, it will be useful to fix the following notation to indicate minors of a matrix.
Definition 2.5 Let G be an m × n matrix with m ≤ n. We denote by G i 1 i 2 ...im the m × m submatrix with columns (i 1 , i 2 , . . . , i m ). We define the minor Definition 2.6 (Horizontal jacobian) Let Ω be an open subset of H n and let x ∈ Ω. The horizontal jacobian of a C 1 mapping f : Ω −→ R k at x is given by From definition of horizontal jacobian, it follows that it only depends on the restriction of g to the horizontal subbundle, namely, from the "sub-Riemannian metric". Let us consider a horizontal frame .
As a consequence, we have the formula be an orthonormal frame with respect to a left invariant metric h and let F : R 2n+1 −→ H n define coordinates with respect to this frame. Then we have F ♯ L 2n+1 = vol 2n+1 , where vol 2n+1 denotes the Riemannian volume measure with respect to the metric h.
Proof. Let A be a measurable set of R 2n+1 . By classical area formula and taking into account the left invariance of both vol p and F ♯ L 2n+1 we have (0)•L = L has jacobian equal to one, then c = 1 and the thesis follows. ✷ Remark 2.8 By previous proposition, the volume measure of a measurable subset A of H n corresponds to the (2n + 1)-dimensional Lebesgue measure of the same subset read with respect to coordinates associated to an orthonormal frame. Here the volume measure is defined by the same left invariant metric.
Recall that (F ♯ µ)(A) = µ(F −1 (A)), where µ is a measure defined on the domain of F and A is a measurable set defined on the codomain of F . The d-dimensional spherical Hausdorff measure S d is defined as where the diameter is considered with respect to a homogeneous distance ρ of H n and we do not consider any dimensional factor. The k-dimensional Hausdorff measure built with respect to the Riemannian distance is denoted by vol k and it corresponds to the classical Riemannian volume measure with respect to the graded metric g, see for instance 3.2.46 of [5].

Definition 2.9 (Horizontal p-vectors)
For each x ∈ H n , we say that any linear combination of wedge products

Definition 2.10 (Vertical p-vectors)
For each x ∈ H n , we say that any linear combination of wedge products For every couple of simple p-vectors v 1 ∧ · · · ∧ v p , w 1 ∧ · · · ∧ w p ∈ Λ p (T x H n ), we define the scalar product induced by the left invariant Riemannian metric g on T x H n as see for instance 1.7.5 of [5] for more details. This allows us to regard the space of vertical p-vectors V p (T x H n ) as the orthogonal complement of the horizontal subspace Λ p (H x H n ). We have the orthogonal decomposition which generalizes the case p = 1, corresponding to We say that ξ V is the vertical projection of ξ and that the mapping We have omitted x in the definition of vertical projection π V .
Definition 2.12 (Characteristic points and transverse points) Let Σ ⊂ Ω be a C 1 submanifold and let x ∈ Σ. We say that x ∈ Σ is a characteristic point if T x Σ ⊂ H x H n and that it is a transverse point otherwise. The characteristic set of Σ is the subset of all characteristic points and it is denoted by C(Σ).
Recall that a tangent p-vector to a p-dimensional submanifold Σ of class C 1 at x ∈ Σ is defined by the wedge product t 1 ∧t 2 ∧· · ·∧t p , where (t 1 , . . . , t p ) is an orthonormal basis of T x Σ. We denote this simple p-vector by τ Σ (x). Notice that the tangent p-vector (which belongs to a one-dimensional space) cannot be continuously defined on all of Σ, unless the submanifold is oriented.
where τ Σ is a tangent p-vector and π V is the vertical projection. The vertical tangent p-vector will be denoted by τ Σ,V (x).

Blow-up at transverse points
This section is devoted to the proof of Theorem 1.1. In the following proposition, we give a simple characterization of characteristic points using vertical tangent p-vectors.
Proof. Let x ∈ Σ and let (t 1 , t 2 , . . . , t p ) be an orthonormal basis of T x Σ. We have the unique decomposition t j = V j + γ j Z, where V j ∈ H x H n for every j = 1, . . . , p. It follows that Assume that x / ∈ C(Σ). If V 1 , V 2 , . . . , V p are linearly dependent, then we get As a result, π V (τ ) = τ hence it is not vanishing. If V 1 , . . . , V p are linearly independent, then all wedge products of the form are non-vanishing for every j = 1, . . . , p. The fact that x is transverse implies that there exists γ j 0 = 0, then the projection is non-vanishing. Conversely, if π V (τ ) = 0, then (20) yields some γ j 1 = 0, therefore Proof. We first notice that Ker df ( This last formula allows us to get our claim as follows. Assume that x ∈ C(Σ).
Then (21) gives From this equation we conclude that df (x) |HxH n is not surjective. Conversely, if df (x) |HxH n is not surjective, then (21) implies therefore T x Σ ⊂ H x H n , namely, x ∈ C(Σ). ✷ Theorem 3.3 Let f : Ω −→ R k be of class C 1 , with surjective differential at each point of Ω. Let Σ denote the submanifold f −1 (0) of Ω and let x ∈ Σ. Then we have Proof. Left invariance of Riemannian metric allows us to consider the left translated submanifold l x −1 Σ. Replacing f with f • l x and Ω with l x −1 Ω we can assume that x is the unit element 0 of H n . Recall that l x : H n −→ H n is the left translation l x (y) = x·y. If x ∈ C(Σ), then Proposition 3.1 and Proposition 3.2 make (22) the trivial identity 0 = 0. Assume that x ∈ Σ \ C(Σ). Then Proposition 3.2 implies that the horizotal gradients span a k-dimensional space of R 2n . Let c 1 , c 2 , . . . , c k ∈ R 2n be orthogonal unit vectors generating this vector space and choose c k+1 , . . . , c 2n ∈ R 2n such that (c 1 , c 2 , . . . , c 2n ) is an orthonormal basis of R 2n . These vectors allow us to define a new horizontal frame We denote by C the 2n × 2n orthogonal matrix whose i-th column corresponds to the vector c i , then by our choice of vectors c j , we obtain where the symbol , denotes the standard scalar product of R 2n . Let us consider F : R 2n+1 −→ H n , defining graded coordinates (y 1 , . . . , y 2n+1 ) associated to the frame (Y 1 , . . . , Y 2n , Z), according to Definition 2.2. Then the differential of f at 0 with respect to (y 1 , . . . , y 2n+1 ) can be represented by the matrix It follows that for everyỹ = (y k+1 , . . . , y 2n+1 ) ∈ A. Then we define the mapping φ : so that differentiating (26) we get for every i = 1, . . . , k and j = k + 1, . . . , 2n + 1. Equations (28) can be more concisely written in matrix form as follows where z = (y 1 , . . . , y k ), the k × k matrix ∇ z f has coefficients f i y l , where i, l = 1, . . . , k and j = k+1, . . . , 2n+1. In order to achieve a more explicit formula for the differential of the implicit map, we explicitly write the inverse matrix of ∇ z f as where C ij (∇ z f ) denotes the cofactor of ∇ z f , which is equal to (−1) i+j det(D ij f ) and D ij f is the (k − 1) × (k − 1) square matrix obtained by removing the i-th row and the j-th column from ∇ z f . In view of (29) we have An elementary formula for computing the determinant of a matrix implies for every j = k + 1, . . . , 2n + 1. As a consequence, we get Note that M 1···k (∇ z f ) corresponds to the determinant of the matrix ∇ z f . As a consequece of (30) and of (25), we conclude that ϕ s y j (0) = 0 for every j = k + 1, . . . , 2n. Previous considerations and expression (27) lead us to the formula where ∇ỹφ(0) is a (2n + 1) × p matrix whose p × p lower block is the identity matrix. Notice that columns of (31) represent a basis of the tangent space T 0 Σ with respect to coordinates (y k+1 , . . . , y 2n+1 ). More precisely, the set of vectors  form an orthonormal basis of T 0 Σ, where we have defined v j = ϕ j y 2n+1 (0). Then the tangent p-vector τ Σ to Σ at 0 is given by the wedge product Obviously, p-vectors Y k+1 (0) ∧ Y k+2 (0) ∧ · · · ∧ Y 2n (0) ∧ Y j (0) are horizontal, hence they disappear in the vertical projection. It follows that , therefore we clearly obtain Due to formula (30) in the case j = 2n + 1 and to (25), we obtain , then (32) shows the valdity of (22) in the case x = 0. Left invariance of the Riemannian metric g leads us to the conclusion. ✷ Definition 3.4 (Metric factor) Let τ be a vertical simple p-vector of Λ p (h n ) and let L(τ ) be the unique associated subspace, with L = exp L(τ ). The metric factor of a homogeneous distance ρ with respect to τ is defined by where F : R 2n+1 −→ H n defines a system of graded coordinates, H p |·| denotes the p-dimensional Hausdorff measure with respect to the Euclidean distance of R 2n+1 and B 1 is the unit ball of H n with respect to the distance ρ. Recall that the subspace associated to a simple p-vector τ is defined as {v ∈ h n | v ∧ τ = 0}.

Remark 3.5
In the case of subspaces L of codimension one, the notion of metric factor fits into the one introduced in [19]. It is easy to observe that the notion of metric factor does not depend on the system of coordinates we are using. In fact, is an Euclidean isometry whenever F 1 , F 2 : R 2n+1 −→ H n represent systems of graded coordinates with respect to the same left invariant Riemannian metric.
From (39), we conclude that To compute J g φ(0), we use both the canonical form of the tangent space T 0 Σ given by (36) and the fact that our frame (Y 1 , . . . , Y 2n , Z) is orthonormal. Thus, according to Definition 2.4 the Riemannian jacobian of φ at zero is given by where we have defined v j = ϕ j y 2n+1 (0) for every j = 1, . . . , 2n. Again, following the same steps of the proof of Theorem 3.3, we get The subspace L(τ Σ,V (x)) associated to the p-vector τ Σ,V (x) satisfies the relation exp L(τ Σ,V (x)) = F (Π) therefore the metric factor of ρ with respect to τ Σ,V (x) is H p (Π ∩B 1 ). This fact along with (45) implies the validity of (1) and ends the proof. ✷

Spherical Hausdorff measure of submanifolds
This section deals with various applications of Theorem 1.2. A key result to obtain this theorem is the S Q−k -negligibility of characteristic points of a k-codimensional submanifold of a Carnot group of Hausdorff dimension Q, see [22]. This result in the case of Heisenberg groups reads as follows.
Theorem 4.1 Let Σ ⊂ Ω be a C 1 submanifold of dimension p. Then the set of characteristic points C(Σ) is S p+1 -negligible.
Remark 4.2 In order to apply the negligibility result of [22] one has to check that the notion of characteristic point in arbitrary stratified groups coincides with our definition stated in the Heisenberg group. According to [22] a point x ∈ Σ is characteristic if If x is characteristic according to Definition 2.12, then Proof. We apply Theorem 2.10.17(2) and Theorem 2.10.18(1) of [5], hence from limit (1) and Theorem 4.1 the proof follows by a standard argument. ✷ Next, we present a class of homogeneous distances in the Heisenberg group which possess constant metric factor. The standard system of graded coordinates F : R 2n+1 −→ H n induced by (X 1 , . . . , X 2n , Z) will be understood in the sequel. To simplify notation we will write x = F (x, x 2n+1 ) ∈ H n , withx = (x 1 , . . . , x 2n ) ∈ R 2n . Proposition 4.5 Let F : R 2n+1 −→ H n define standard coordintates and let ρ be a homogeneous distance of H n such that ρ(0, F (·)) : R 2n+1 −→ R only depends on (|x|, x 2n+1 ). Then θ ρ p (τ ) = θ ρ p (τ ) whenever τ,τ are vertical simple p-vectors.
Due to Proposition 4.5, both of these distances have constant metric factor.

Remark 4.7
The metric factor depends on the Riemannian metric g we have fixed. Furthermore, if we divide it by the volume of the unit ball B 1 (or any other fixed subset of positive measure), then we obtain a number only depending on the restriction of the Riemannian metric to HH n .
Lemma 4.8 Letg be a left invariant Riemannian metric such thatg |HH n = g |HH n .
Theorem 4.9 Let Σ be a p-dimensional submanifold of Ω and letg a left invariant metric such thatg |HH n = g |HH n . Then The previous theorem is an immediate conseqauence of Corollary 4.3 and Lemma 4.8. Next, we apply (2) to compute the spherical Hausdorff measure of some submanifolds. We will use the following proposition.
for every measurable set A ⊂ H n , where the norm | · | is induced by the scalar product (17) on p-vectors.
Remark 4.14 It is curious to notice that the density of S 3 H 1 restricted to the paraboloid P, computed in (61), is proportional to the density of S 3 H 1 restricted to the horizontal projection of P onto the plane F ({(x 1 , x 2 , x 3 ) | x 3 = 0}) ⊂ H 1 , whose density is given by (60) in the case a 1 = a 2 = 1 and b = c = 0. We have then defining the submanifold L = Φ([α, β]), the formula S 2 H 1 (L) = |b| (β − α) holds.
Another consequence of (2) is the lower semicontinuity of the spherical Hausdorff measure with respect to weak convergence of regular currents. To see this, it suffices to establish the following formula where F p c (Ω) is the space of smooth p-forms with compact support in Ω with |ω| ≤ 1. The norm of ω is defined making the standard frame of p-forms (dx 1 , dx 2 , . . . , dx 2n ,θ) orthonormal and extending this scalar product to p-forms exactly as we have seen in formula (17). The 1-formθ is the so called contact form written in standard coordinates. Note that (dx 1 , dx 2 , . . . , dx 2n ,θ) is the dual basis of (X 1 , . . . ,X 2n ,Z). Formula (63) follows from (2) observing that Σ |τ Σ,V | dvol p = sup ω∈F p c (Ω) Σ τ Σ,V , ω dvol p , as one can check by standard arguments. As a consequence of these observations, we can establish the following proposition.
for every ω ∈ F p c (Ω). Then (63) ends the proof. ✷ Remark 4.17 It is clear the importance of (66) in studying versions of the Plateau problem with respect to the geometry of Heisenberg groups.
Recall that the horizontal normal is the orthogonal projection of the normal to ν(x) to T x Σ onto the horizontal subspace H x H n . In the next proposition we show that in codimension one an explicit relationship can be established between vertical tangent 2n-vector and horizontal normal ν H . where ν H = 2n j=1 ν j H X j and τ Σ,V = 2n j=1 τ j Σ,V X 1 ∧ · · · X j−1 ∧ X j+1 ∧ · · · X 2n ∧ Z. In particular, the equality |τ Σ,V | = |ν H | holds.

Coarea formula
This section is devoted to the proof of Theorem 1.3. Next, we recall the Riemannian coarea formula, see Section 13.4 of [4].
Theorem 5.1 Let f : H n −→ R k be a Riemannian Lipschitz function, with 1 ≤ k < 2n + 1. Then for any summable map u : H n −→ R, the following formula holds where p = 2n + 1 − k In the previous theorem the Heisenberg group H n is equipped with its left invariant Riemannian metric g. The terminology "Riemannian Lipschitz map" means that the map is Lipschitz with respect to the Riemannian distance.
Proof of Theorem 1.3. We first prove (4) in the case f is defined on all of H n and is of class C 1 . Let Ω be an open subset of H n . In view of Riemannian coarea formula (71), we have where u : Ω −→ [0, +∞] is a measurable function. Note that in the left hand side of (72) we have used the Lebesgue measure in that, by Proposition 2.7, it coincides with the volume measure expressed in terms of standard coordinates, namely F ♯ (L 2n+1 ) = vol 2n+1 . Now we define and use (72), obtaining In fact, due to the validity of (76) for C 1 mappings, we have Remark 5.2 Notice that once f : A −→ R k in the previous theorem is considered with respect to standard coordinates it is easy to check that the locally Lipschitz property with respect to the Euclidean distance of R 2n+1 is equivalent to the locally Lipschitz property with respect to the Riemannian distance.