Area of intrinsic graphs and coarea formula in Carnot Groups

We consider submanifolds of sub-Riemannian Carnot groups with intrinsic $C^1$ regularity ($C^1_H$). Our first main result is an area formula for $C^1_H$ intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $C^1_H$ submanifolds into level sets of a $C^1_H$ function.


Introduction
The interest towards Analysis and Geometry in Metric Spaces grew drastically in the last decades: a major effort has been devoted to the development of analytical tools for the study of geometric problems, and sub-Riemannian Geometry provided a particularly fruitful setting for these investigations. The present paper aims at giving a contribution in this direction by providing some geometric integration formulae, namely: an area formula for submanifolds with (intrinsic) C 1 regularity, and a coarea formula for slicing such submanifolds into level sets of maps with (intrinsic) C 1 regularity.
We will work in the setting of a Carnot group G, i.e., a connected, simply connected and nilpotent Lie group with stratified Lie algebra. We refer to Section 2.1 for precise definitions; here, we only recall that Carnot groups have a distinguished role in sub-Riemannian Geometry, as they provide the infinitesimal models (tangents spaces) of sub-Riemannian manifolds, see e.g. [4]. As usual, a Carnot group is endowed with a distance ρ that is left-invariant and 1-homogeneous with respect to the group dilations.
Our main objects of investigation are C 1 H submanifolds, which are introduced as (noncritical) level sets of functions with intrinsic C 1 regularity: let us briefly introduce the relevant definitions, which are more precisely stated in Section 2. Given an open set Ω ⊂ G and another Carnot 1 group G ′ , a map f : Ω → G ′ is said to be of class C 1 H if it is differentiable à la P. Pansu [41] at all p ∈ Ω and the differential D H f p : G → G ′ is continuous in p. Let us mention that the C 1 H regularity of f is equivalent to its strict Pansu differentiability (see Proposition 2.4): such a notion is introduced in Section 2.3 and turns out to be useful for simplifying several arguments. Given a Carnot group G ′ , a set Σ ⊂ G is a C 1 H (G; G ′ )-submanifold if it is locally a level set of a map f : G → G ′ of class C 1 H such that, at all points p, D H f p is surjective and ker D H f p splits G. We say that a normal homogeneous subgroup W < G splits G if there exists another homogeneous subgroup V < G, which is complementary to W, i.e., such that V ∩ W = {0} and G = WV. Observe that V is necessarily isomorphic to G ′ , see Remark 2.8. We will also say that p is split-regular for f if D H f p is surjective and ker D H f p splits G.
In Sections 2.4 and 2.5 we prove that an Implicit Function Theorem holds for a C 1 H submanifold Σ; namely, Σ is (locally) an intrinsic graph, i.e., there exist complementary homogeneous subgroups W, V of G and a function φ : A → V defined on an open subset A ⊂ W such that Σ coincides with the intrinsic graph {wφ(w) : w ∈ A} of φ. The function φ is of class C 1 W,V (see Definition 2.13) and it turns out to be intrinsic Lipschitz continuous according to the theory developed in recent years by B. Franchi, R. Serapioni and F. Serra Cassano, see e.g. [15,17,18]. We have to mention that both the Implicit Function Theorem and the intrinsic Lipschitz continuity of φ follow also from [34,Theorem 1.4]: the proofs we provide in Sections 2.4-2.5, however, seem shorter than those in [34] and allow for some finer results we need, see e.g. Lemmas 2.12 and 2.14. For related results, see [3,5,14,16,44].
Our first main result is an area formula for intrinsic graphs of class C 1 W,V (hence, in particular, for C 1 H submanifolds) where complementary subgroups W < G and V < G are fixed with W normal. Throughout the paper we denote by ψ d either the spherical or the Hausdorff measure of dimension d in G. The function A( · ) appearing in (1) is continuous and it is called area factor: it is defined in Lemma 3.2 and it depends only on (W, V and) the homogeneous tangent space T H p Σ at points p ∈ Σ. The definition of area factor in Lemma 3.2 is only implicit, but of course we expect it can be made more explicit in terms of suitable derivatives of the map φ: to the best of our knowledge, this program has been completed only in Heisenberg groups, see e.g. [2,3,6,7,16]. A relevant tool in the proof of Theorem 1.1 is a differentiation theorem for measures (Proposition 2.2) which is based on the so-called Federer density (9): the importance of this notion was pointed out only recently by V. Magnani, see [35,36,37] and [19]. Observe that the validity of a (currently unavailable) Rademacher-type Theorem for intrinsic Lipschitz graphs would likely allow to extend Theorem 1.1 to the case of intrinsic Lipschitz φ.
A first interesting consequence of Theorem 1.1 is the following Corollary 1.2, which is reminiscent of the well-known equality between Hausdorff and spherical Hausdorff measures on C 1 submanifolds (and, more generally, on rectifiable subsets) of R n . We refer to Definitions 2.18 and 2.19 for the notions of countably (G; G ′ )-rectifiable set R ⊂ G and of approximate tangent space T H R. Such sets have Hausdorff dimension Q − m, where Q and m denote, respectively, the homogeneous dimensions of G, G ′ ; we write H Q−m , S Q−m , respectively, for Hausdorff and spherical Hausdorff measures. We denote by T G,G ′ the space of possible tangent subgroups to (G; G ′ )-rectifiable sets 2 and, by abuse of notation, we write T H R for the map R ∋ p → T H p R ∈ T G,G ′ . Moreover, if G is a Heisenberg group H n with a rotationally invariant distance ρ and G ′ = R, then the function a is constant, i.e., there exists C ∈ [1, 2 2n+1 ] such that Heisenberg groups and rotationally invariant distances are defined in Section 2.1 by condition (34), while Corollary 1.2 is proved in Section 3. To the best of our knowledge, this result is new even in the first Heisenberg group H 1 , see also [37, page 359]. Corollary 1.2 is deeply connected to the isodiametric problem, see Remark 3.3.
Not unrelated with Corollary 1.2 is another interesting consequence of Theorem 1.1, namely, the existence of the density of Hausdorff and spherical measures on rectifiable sets. In Corollary 3. 6 we indeed prove that, if R ⊂ G is (G; G ′ )rectifiable, then the limit d(p) := lim r→0 + ψ Q−m (R ∩ U(p, r)) r Q−m exists for ψ Q−m -a.e. p ∈ R, where U(p, r) is the open ball of center p and radius r for the distance of G. Actually, d(p) depends only on T H p R, in a continuous way. When G is the Heisenberg group H n endowed with a rotationally invariant distance, G ′ = R m for some 1 ≤ m ≤ n, and ψ is the spherical measure, then d is constant, see Corollary 5.5.
The area formula is a key tool also in the proof of our second main result, the coarea formula in Theorem 1.3 below. The classical coarea formula was first proved in the seminal paper [12] and it is one of the milestones of Geometric Measure Theory. Sub-Riemannian coarea formulae have been obtained in [29,30,31,32,22,23], assuming classical (Euclidean) regularity on the slicing function u, and in [33,38,39], assuming intrinsic regularity but only in the setting of the Heisenberg group. Here we try to work in the utmost generality: we consider a C 1 H submanifold Σ ⊂ G, seen as the level set of a C 1 H map f with values in a homogeneous group M, and we slice it into level sets of a map u with values into another homogeneous group L. We assume for the sake of generality (see below) that L, M are complementary subgroups of a larger homogeneous group K = LM; we also denote by Q, m, ℓ the homogeneous dimensions of G, M, L, respectively. and assume that all points in Ω are split-regular for f , so that Σ := {p ∈ Ω : f (p) = 0} is a C 1 H submanifold. Consider a function u : Ω → L such that uf ∈ C 1 H (Ω; K) and assume that Then, for every Borel function h : Σ → [0, +∞) the equality holds.
In (4), the symbol C(T H p Σ, D H (uf ) p ) denotes the coarea factor: let us stress that it depends only on the restriction of u to Σ and that it does not depend on the choice of f outside of Σ, see Remark 4.2. The ψ ℓ -measurability of the function The assumption uf ∈ C 1 H (Ω; K) becomes more transparent when K = L × M is a direct product (roughly speaking, when L, M are "unrelated" groups): in this case, it is in fact equivalent to the C 1 H regularity of u. Moreover, since Eventually, the statement of Theorem 1.3 can at the same time be simplified, stated in a more natural way, and generalized to rectifiable sets, as follows. holds.
Remark 1.5. Let us stress that assumptions (3) and (5) cannot be easily relaxed: given a map u ∈ C 1 H (Ω, R 2 ) defined on an open subset Ω of the first Heisenberg group H 1 ≡ R 3 , the validity of a coarea formula of the type is indeed a challenging open problem as soon as D H u p is surjective, see e.g. [24,26,38]. In our notation, this situation corresponds to M = {0} and L = R 2 . Since the kernel of any homogeneous surjective morphism H 1 → R 2 is the center of H 1 , which does not admit any complementary subgroup, no point can be split-regular for u. Therefore, if (5) holds, then C(D H u p ) = 0 by Proposition 4.5, and thus both sides of the coarea formula are null. In particular, (5) implies that for L 2 -a.e. s ∈ R 2 , ψ 2 (Ω ∩ u −1 (s)) = 0. However, a coarea formula was proved for u : H n → R 2n , assuming u to be of class C 1,α H , see [24,Theorem 6.2.5] and also [38,Theorem 8.2]. Remark 1.6. The following weak version of Sard's Theorem holds: under the assumptions and notation of Theorem 1.3, then Moreover, since every level set Σ ∩ u −1 (s) is a C 1 H submanifold around split-regular points of uf , Theorem 1.3 implies that Clearly, statements analogous to (6) and (7) hold under the assumptions and notation of either Corollary 1.4 or Theorem 1.7 below.
The proof of Theorem 1.3 follows the strategy used in [12] (see also [33]) and, as already mentioned, it stems from the area formula of Theorem 1.1, as we now describe. First, in Proposition 4.4 we prove a coarea inequality, that in turn is based on an "abstract" coarea inequality (Lemma 4.3) for Lipschitz maps between metric spaces. Second, in Lemma 4.5 we prove Theorem 1.3 in the "linearized" case when both f and u are homogeneous group morphisms: in this case formula (4) holds with a constant coarea factor C(P, L) which depends only on the normal homogeneous subgroup P := ker f and on the homogeneous morphism L = u (actually, on L| Σ only). Lemma 4.5, whose proof is a simple application of Theorem 1.1, actually defines the coarea factor C(P, L). The proof of Theorem 1.3 is then a direct consequence of Theorem 4.1, which states that for ψ Q−m -a.e. p ∈ Σ the Federer density Θ ψ d (µ Σ,u ; p) of the measure For "good" points p, i.e., when D H (uf ) p | T H p Σ is onto L, such equality is obtained by another application of Theorem 1.1, see Proposition 4.7: this is the point where one needs the assumption (3), which guarantees that, locally around good points, the level sets Σ ∩ u −1 (s) are C 1 H submanifolds. The remaining "bad" points, where D H (uf ) p | T H p Σ is not surjective on L, can be treated using the coarea inequality, see Lemma 4.8. Recall that the classical Euclidean coarea formula is proved when the slicing function u is only Lipschitz continuous. Extending Theorem 1.3 to the case where u : Σ → L is only Lipschitz seems for the moment out of reach. Observe that one should first provide, for a.e. p ∈ Σ, a notion of Pansu differential of u on T H p Σ: this does not follow from Pansu's Theorem [41]. Furthermore, the function f in Theorem 1.3 should play no role, and actually any result should depend only on the restriction of u to Σ.
Let us also stress that, to the best of our knowledge, Theorem 1.3 provides the first sub-Riemannian coarea formula that is proved when the set Σ is not a positive ψ Q -measure subset of G (i.e., in the notation of Theorem 1.3, when M = {0}). The only exception to this is [39,Theorem 1.5], where a coarea formula was proved for C 1 H submanifolds of codimension 1 in Heisenberg groups H n , n ≥ 2. As a corollary of Theorem 1.3, we are able both to extend this result, to all codimensions not greater than n, and to improve it, in the sense that we show that the implicit "perimeter" measures considered in [39, Theorem 1.5] on the level sets of u are indeed Hausdorff or spherical measures. Furthermore, when H n is endowed with a rotationally invariant distance, u takes values in R ℓ , and the measures ψ d under consideration are S d , then the coarea factor coincides up to constants with the quantity In (8), the point p belongs to a rectifiable set R ⊂ H n and, by abuse of notation, we use standard exponential coordinates on H n ≡ R 2n+1 to identify T H p R with a (2n + 1 − m)-dimensional plane; with this identification D H u p is a linear map on R 2n+1 that is, actually, independent of the last "vertical" coordinate. The superscript T denotes transposition. Theorem 1.7 (Coarea formula in Heisenberg groups). Consider an open set Ω ⊂ H n , a (H n , R m )-rectifiable set R ⊂ Ω and a function u ∈ C 1 H (Ω; R ℓ ) such that 1 ≤ m + ℓ ≤ n. Then, for every Borel function h : holds. Moreover, if H n is endowed with a rotationally invariant distance ρ, then there exists a constant c = c(n, m, ℓ, ρ) > 0 such that The first statement of Theorem 1.7 is an immediate application of Corollary 1.4, while the second one needs an explicit representation for the spherical measure on vertical subgroups of H n (i.e., elements of T H n ,R k ) which use results of [7]. See Proposition 5.1.
Acknowledgements. The authors are grateful to F. Corni, V. Magnani, R. Monti and P. Pansu for several stimulating discussions. They wish to express their gratitude to A. Merlo for suggesting to address the density existence problem of Corollary 3.6.

Preliminaries
2.1. First definitions. Let V be a real vector space with finite dimension and [·, ·] : V × V → V be the Lie bracket of a Lie algebra g = (V, [·, ·]). We say that g is graded if subspaces V 1 , . . . , V s are fixed so that where we agree that V k = {0} if k > s. Graded Lie algebras are nilpotent. A graded Lie algebra is stratified of step s if equality [V 1 , V j ] = V j+1 holds and V s = {0}. Our main object of study are stratified Lie algebras, but we will often work with subspaces that are only graded Lie algebras.
On the vector space V we define a group operation via the Baker-Campbell-Hausdorff formula where [p r1 q s1 p r2 q s2 · · · p rn q sn ] = [p, [p, . . . , The sum in the formula above is finite because g is nilpotent. The resulting Lie group, which we denote by G, is nilpotent and simply connected; we will call it graded group or stratified group, depending on the type of grading of the Lie algebra. The identification G = V = g corresponds to the identification between Lie algebra and Lie group via the exponential map exp : g → G. Notice that p −1 = −p for every p ∈ G and that 0 is the neutral element of G.
If g ′ is another graded Lie algebra with underlying vector space V ′ and Lie group G ′ , then, with the same identifications as above, a map V → V ′ is a Lie algebra morphism if and only if it is a Lie group morphism, and all such maps are linear. In particular, we denote by Hom h (G; G ′ ) the space of all homogeneous morphisms from G to G ′ , that is, all linear maps V → V ′ that are Lie algebra morphisms (equivalently, Lie group morphisms) and that map V j to V ′ j . If g is stratified, then homogeneous morphisms are uniquely determined by their restriction to V 1 .
For λ > 0, define the dilations as the maps δ λ : Notice that δ λ δ µ = δ λµ and that δ λ ∈ Hom h (G; G), for all λ, µ > 0. Notice also that a Lie group morphism F : We say that a subset M of V is homogeneous if δ λ (M ) = M for all λ > 0. Let P be a homogeneous subgroup of G and θ a Haar measure on P. Since δ λ | P is an automorphism of P, there is c λ > 0 such that (δ λ ) # θ = c λ θ. Since the map λ → δ λ | P is a multiplicative one-parameter group of automorphisms, the map λ → c λ is a continuous automorphism of the multiplicative group (0, +∞), hence c λ = λ −d for some d ∈ R. As δ λ is contractive for λ < 1, we actually have d > 0. Since any other Haar measure of P is a positive multiple of θ, the constant d does not depend on the choice of the Haar measure. We call such exponent d the homogeneous dimension of P. The homogeneous dimension of the ambient space G is denoted by Q and it is easy to see that Q := s i=1 i dim V i . A homogeneous distance on G is a distance function ρ that is left-invariant and 1-homogeneous with respect to dilations, i.e., (i) ρ(gx, gy) = ρ(x, y) for all g, x, y ∈ G; (ii) ρ(δ λ x, δ λ y) = λρ(x, y) for all x, y ∈ G and all λ > 0.
When a stratified group G is endowed with a homogeneous distance ρ, we call the metric Lie group (G, ρ) a Carnot group. Homogeneous distances induce the topology of G, see [25,Proposition 2.26], and are biLipschitz equivalent to each other. Every homogeneous distance defines a homogeneous norm · ρ : G → [0, +∞), p ρ := ρ(0, p). We denote by |·| the Euclidean norm in R ℓ . The following property relating norm and conjugation, proved in [18, Lemma 2.13], will be useful: there exists C = C(G, ρ) > 0 such that Open balls with respect to ρ are denoted by U ρ (x, r), closed balls by B ρ (x, r), or simply U(x, r) and B(x, r) if it is clear which distance we are using. We also use the notation B(E, r) := {x : d(x, E) ≤ r} for subsets E of G. The diameter of a set with respect to ρ is denoted by diam(E) or diam ρ (E). Notice that diam ρ (U ρ (p, r)) = 2r, for all p ∈ G and r > 0. By left-invariance of ρ it suffices to prove this for p = 0. On the one hand the triangle inequality implies diam ρ (U ρ (0, r)) ≤ 2r.
On the other hand, if v ∈ V 1 is such that ρ(0, v) = r, then ρ(0, v −1 ) = r and If ρ and ρ ′ are homogeneous distances on G and G ′ , the distance between two homomorphisms L, M ∈ Hom h (G; G ′ ) is The function d ρ,ρ ′ is a distance on Hom h (G; G ′ ) inducing the manifold topology.

2.2.
Measures and Federer density. In the following, the word measure will stand for outer measure. We work on G and its subsets endowed with the metric ρ. In particular, the balls are those defined by ρ and the Hausdorff dimension of (G, ρ) coincides with the homogeneous dimension Q.
It is clear that, in the definition of H d , one can ask the covering sets E j to be closed.
Note that contrarily to the usual Euclidean or Riemannian definition, we do not introduce normalization constants; this is due to the fact that the appropriate constant is usually linked to the solution to the isodiametric problem, which is open in Carnot Groups and their subgroups and also highly dependent on the metric ρ. See also Remark 3.3. In the following, ψ d will be either H d or S d and E will be, respectively, the collection of closed subsets of G of positive diameter or the collection of closed balls in G with positive diameter.
If µ is a measure on G, define the ψ d -density of µ at x ∈ G as This upper density is sometimes called Federer density [19,35,36]; note that if ψ d is the spherical measure, its Federer density can differ from the usual spherical density, as the latter involves centered balls. Recall that a measure ν is Borel regular if open sets are measurable and for every A ⊂ G there exists a Borel set A ′ ⊂ G such that A ⊂ A ′ and ν(A ′ ) = ν(A). We will use the following density estimates, which follow from [13, Theorems 2.10.17 and 2.10.18].
Theorem 2.1 (Density estimates). Let ψ d be as above, µ a Borel regular measure, and fix t > 0 and a set A in G. Then A direct consequence of these results is the following (see also [35,Theorem 9] and [19,Theorem 1.11]).

Proposition 2.2. If µ is locally finite and Borel regular on G, and if
is a Borel function which is positive and finite µ-almost everywhere, then Proving that the Federer density is a ψ d -measurable or a Borel function is in general not an easy task; we provide a criterion, which will be useful later in Sections 4.4 and 4.5. Recall that a Borel measure ν is doubling if there exists C ≥ 1 such that ν(U(p, 2r)) ≤ C ν(U(p, r)) for all p ∈ G and r > 0. Proposition 2.3. Given a set Σ ⊂ G such that ψ d Σ is locally doubling Borel regular measure, assume that µ is a locally finite Borel regular measure, absolutely continuous with respect to ψ d Σ; then Proof. It is well-known (see e.g. [43]) that Radon-Nikodym Differentiation Theorem holds for differentiating a measure with respect to a doubling measure. Precisely, by combining [43, Theorems 2.2, 2.3, 3.1] one infers that the Radon-Nikodym derivative As a consequence, we have only to prove that Θ ψ d (µ; p) = Θ(p) for ψ d -a.e. p ∈ Σ.
In turn, it is enough to show that, for every fixed s, t ∈ Q, s < t, the sets where the last inequality is a consequence of Theorem 2.1 (ii). Therefore, where the last inequality is a consequence of Theorem 2.1 (i). Therefore ψ d (B) = 0.
2.3. Pansu differential. Let G and G ′ be two Carnot groups and The map L is called Pansu differential of f at p and it is denoted by Clearly, in this case f is Pansu differentiable at p and L = D H f (p). The next results allows us to simplify several arguments in the sequel: Proof. Assume that f ∈ C 1 H (Ω, G ′ ) and let p ∈ Ω be fixed; then, by [34, Theorem and the strict differentiability of f at p follows.
Conversely, assume that f is strictly Pansu differentiable at all points in Ω; we have to prove that p → D H f p is continuous. Assume not, i.e., assume there exist δ > 0 and, for every n ∈ N, points x n ∈ Ω and v n ∈ G such that v n ρ = 1, x n → p and By strict differentiability of f at p there existn ands > 0 such that In particular, for every n ≥n and s ∈ (0,s) we have This would contradict the differentiability of f at x n .
Proof. Let p ∈ Ω. By strict differentiability of f at p, there is ε > 0 such that Remark 2.6. The notion of Pansu differentiability, as well as Lemma 2.5, can be stated also when the target group G ′ is only graded. However, there is no loss of generality in assuming G ′ to be stratified. Indeed, if f : Ω → G ′ is locally Lipschitz, then the image of a rectifiable curve in G is a rectifiable curve in G ′ tangent to the first layer V ′ 1 in the grading of G ′ ; since G is stratified, each connected component U of Ω is pathwise connected by rectifiable curves, and this implies that f (U ) is contained in (a coset of) the stratified subgroup of G ′ generated by V ′ 1 . Moreover, as soon as f is open, or has a regular point, then G ′ must be a Carnot group.
2.4. Intrinsic graphs and implicit function theorem. We refer to [18] for a more general theory of intrinsic graphs. Recall the identification G = g = V that we made in Section 2.1.
A homogeneous subgroup W is complementary to a homogeneous subgroup V if G = WV and W ∩ V = {0}. We denote by W V the set of all homogeneous subgroups of G that are complementary to V. By Lemma 2.7, we have W ∈ W V if and only if V ∈ W W . Again by Lemma 2.7, any choice of V and W ∈ W V gives two projections (10) π W : G → W, which are defined, for every p ∈ G, by requiring π W (p) = w ∈ W and π V (p) = v ∈ V to be the only elements such that p = wv. We will also write p W and p V for π W (p) and π V (p), respectively. We say that a normal homogeneous subgroup W splits G if W W = ∅. In this case we call a choice of W and V ∈ W W a splitting of G and we write G = W · V. We say that p ∈ Ω is a split-regular point of f if the Pansu differential of f at p exists and is surjective, and if ker(D H f (p)) splits G. Recall that the kernel of a group morphism is always normal. A singular point is a point that is not split-regular.
Notice that a point can fail to be split-regular for f ∈ C 1 H (Ω; G ′ ) for two distinct reasons: non-surjectivity of the differential, or non-existence of a splitting of G with the kernel of D H f p at some point p. However, the set of split-regular points is open, i.e., if D H f p is surjective and (ker D H f p ) · V is a splitting, then, for q close enough to p, D H f q is surjective and (ker D H f q ) · V is a splitting.
∈ Ω is a split-regular point and V is complementary to ker(D H f (p)), then there are a neighborhood U of p and C > 0 such that, for all q ∈ U and v ∈ V with qv ∈ U , Proof. Arguing by contradiction, assume that there are sequences q j ∈ Ω and v j ∈ V\{0} such that q j → p and v j → 0 as j → ∞, and ρ ′ (f (q j ), f (q j v j )) ≤ v j ρ /j. Up to passing to a subsequence, we can assume that there existsw = lim j→∞ δ vj −1 v j . It follows thatw ∈ V and w ρ = 1. Moreover, by strict differentiability The proof of the following lemma is inspired by [ Proof of Lemma 2.10. First, we prove that there is an open neighborhood U ⊂ Ω 0 of o such that the restriction g| pV : pV ∩ U → G ′ is injective, for all p ∈ U . Arguing by contradiction, suppose that this is not the case. Then there are sequences p j , q j → o such that p −1 j q j ∈ V and g(p j ) = g(q j ). From the strict Pansu differentiability of g at o, it follows that By the compactness of the sphere {v ∈ V : v ρ = 1}, up to passing to a subsequence, there is v ∈ V with v ρ = 1 such that lim j→∞ δ ρ(pj ,qj ) −1 (p −1 j q j ) = v. Therefore, we obtain D H g(o)v = 0, in contradiction with the assumptions. This proves the first claim.
Second, since the restriction g| pV∩U : pV ∩ U → G ′ is a continuous and injective map, and since both V and G ′ are topological manifolds of the same dimension, then we can apply the Invariance of Domain Theorem and obtain that g| pV∩U : We claim that there is A ⊂ W open such that π W (o) ∈ A and such that for every p ∈ oV ∩ U 2 and for every a ∈ A there is q ∈ aV∩U 1 such that g(p) = g(q). Arguing by contradiction, suppose that this is not the case. Then there are sequences a j ∈ W with a j → π W (o) and . By the compactness ofŪ 2 and the continuity of g, for each j there is q j ∈ a j V ∩Ū 1 such that Since g is a homeomorphism on each fiber pV ∩ U and since g( Up to passing to a subsequence, there are p 0 ∈ oV ∩Ū 2 and q 0 ∈ oV∩∂U 1 such that p j → p 0 and q j → q 0 . Now, notice that a j π W (o) −1 → 0 and that, for j large enough, we have a j π W (o) −1 p j ∈ a j V ∩ U 1 . Therefore, using (11), that is, g(p 0 ) = g(q 0 ). Since p 0 ∈ oV ∩Ū 2 and q 0 ∈ oV ∩ (U \ U 1 ), this contradicts the injectivity of g on oV ∩ U and proves the claim.
Next, let B := g(oV ∩ U 2 ), which is an open neighborhood of g(o), and Ω := The previous claims imply that for every a ∈ A and every b ∈ B there is a unique v ∈ V such that av ∈ Ω and g(av Finally, we claim that the map Φ(a, b) := aϕ(a, b) is a homeomorphism A × B → Ω. Notice that, if p = Φ(a, b), then a = π W (p) and b = g(p): therefore, Φ is injective. Moreover, if p ∈ Ω, then π W (p) ∈ A, g(p) ∈ B and Φ(π W (p), g(p)) = p: therefore, Φ is also surjective. Finally, since Φ −1 : Ω → A × B is a continuous bijection, then it is a homeomorphism by the Invariance of Domain Theorem. This completes the proof.
We observe that, when g : G → G ′ is a homogeneous group morphism, then the statement of Lemma 2.10 holds with A = W, B = G ′ and Ω = G. Lemma 2.12. Under the assumptions and notation of Lemma 2.10, suppose o = 0 and define for λ > 0 Let ϕ 0 be the implicit function associated with D H g(0), that is, Then ϕ λ → ϕ 0 locally uniformly as λ → 0 + .
Observe that, since V is isomorphic to G ′ , the homogeneous dimension of W is equal to that of Σ.  An important property of the parametrizing map φ is that it is intrinsic Lipschitz in accordance with the theory developed by B. Franchi, R. Serapioni and F. Serra Cassano, see e.g. [15,18]. We recall that, given a splitting G = W · V and A ⊂ W, a map φ : A → V is intrinsic Lipschitz if there exists C ⊂ G such that the following conditions hold (a) C is a cone i.e., δ λ C = C for all λ ≥ 0; Let Ω be a neighborhood of o with f ∈ C 1 H (Ω; G ′ ) such that Σ ∩ Ω = {p ∈ Ω : f (p) = f (o)} and all points in Ω are split-regular for f . Up to shrinking Ω, we can also assume, by Lemma 2.9, that there exists C > 0 such that and that, by Lemma 2.5, f : Requirements (a) and (b) above are clearly satisfied; to prove (c), let p ∈ Σ ∩ Ω and q ∈ (Σ ∩ Ω) ∩ C. Then there exists v ∈ V such that ρ(q, pv) < C L v ρ , hence We conclude that f (q) = f (p) and thus q / ∈ Σ. This completes the proof.
The following result is an easy consequence of Lemma 2.14, Corollary 2.16 and [18, Theorem 3.9]. We denote by ψ d either the d-dimensional Hausdorff or d-dimensional spherical Hausdorff measure on G as in Section 2.2.
Proposition 2.17 (Local Ahlfors regularity of the surface measure on C 1 H submanifolds). Let Σ ⊂ G be a C 1 H and let d := dim H Σ; then, for every compact set K ⊂ Σ there exists C = C(K) > 0 such that In particular, the measure ψ d Σ is locally doubling.
Some of the results of this paper hold for the more general class of rectifiable sets that we now introduce. Definition 2.18 (Rectifiable sets). We say that a set R ⊂ G is countably (G; G ′ )rectifiable if there exists G ′ and countably many C 1 H (G; G ′ )-submanifolds Σ j ⊂ G, j ∈ N, such that, denoting by Q, m the homogeneous dimensions of G, G ′ , one has We say that R is (G; G ′ )-rectifiable if, moreover, ψ Q−m (R) < +∞.
The groups G, G ′ will be usually understood and we will simply write rectifiable in place of (G; G ′ )-rectifiable. Notice that, by Remark 2.8, if ψ Q−m (R) > 0, then the group G ′ is uniquely determined by R up to biLipschitz isomorphism. We recall also that this notion of rectifiability is not known to be equivalent to the ones by means of cones, as in [15,18,8,21].
A key object in the theory of rectifiable sets is the approximate tangent space.
Definition 2.19 (Approximate tangent space). Let R ⊂ G be countably (G; G ′ )rectifiable and let Σ j , j ∈ N, be as in Definition 2.18; for every ψ Q−m -a.e. p ∈ R we define the approximate tangent space T H p R to R at p as Definition 2.19 is well-posed provided one shows that, for ψ Q−m -a.e. p ∈ R, T H p R does not change if in Definition 2.18 one changes the covering family of submanifolds defined on open sets Ω ′ , Ω ′′ ⊂ G and all points are split-regular for f ′ , f ′′ , then (see also [10, Section 2]) Let I be the set in (13). Assume by contradiction that ψ Q−m (I) > 0; we can without loss of generality suppose that Σ ′ is the intrinsic graph of a map φ : A → V defined on an open set A ⊂ W for some splitting G = W · V. Let J := {w ∈ A : wφ(w) ∈ I}; by Theorem 1.1 one has ψ Q−m (J) > 0, hence there existsw ∈ J such that lim r→0 + ψ Q−m (J ∩ U(w, r)) ψ Q−m (W ∩ U(w, r)) = 1.
Taking Lemma 2.14 (iii) into account, it is then a routine task to prove that the blow-up of I atp :=wφ(w), i.e., the limit lim λ→0 + δ 1/λ (p −1 I) in the sense of local Hausdorff convergence, is T H p Σ ′ . This implies that T H p Σ ′ ⊃ T H p Σ ′′ and in turn, by equality of the dimensions, that T H p Σ ′ = T H p Σ ′′ : this is a contradiction.

The area formula
Let P be a homogeneous subgroup of G with homogeneous dimension d and let θ be a Haar measure on P. By dilation invariance of E and P one has This simple observation turns out to be useful to study the Federer density Θ ψ d of ψ d P.
Lemma 3.1. Let P be a homogeneous subgroup of G with homogeneous dimension d and let ψ d be either the spherical or the Hausdorff d-dimensional measure on G. Then ψ d P is a Haar measure on P and for all x ∈ P, Proof. As E and ρ are left invariant, ψ d P is a left invariant measure on P. Therefore, we only need to show that it is non zero and locally finite to prove that it is a Haar measure. Fix a Haar measure θ on P. Since θ is d-homogeneous, θ is Ahlfors d-regular on (P, ρ), therefore there are constants 0 < c < C such that for all Borel subsets B ⊂ P, see for instance [20,Exercise 8.11]. By basic comparisons of the Hausdorff and spherical measures, we infer that ψ d is non zero and locally finite. We can conclude that ψ d is a Haar measure on P.
It remains to prove the equalities in (15). The first equality now follows from (14) and left-invariance. The second equality follows instead from Theorem 2.1.
The following lemma proves Theorem 1.1 in a "linearized" case and allows to define the area factor A. Lemma 3.2 (Definition of the area factor). Let W · V be a splitting of G with W normal. Assume that P is a homogeneous subgroup of G which is also an intrinsic graph W → V and let Φ P : W → P be the corresponding graph map. Then, there exists a positive constant A(P), which we call area factor, such that Furthermore, the area factor is continuous in P.
Proof. In order to prove the first part of the lemma it suffices to show that µ := Φ P# (ψ d W) is a Haar measure on P. To see that it is locally finite, note that Φ P is a homeomorphism between W and P and that therefore bounded open sets in P have finite positive µ measure. We need to prove that µ is left invariant. Choose a set E ⊂ P. Let p = p W p V be a point on P and pick a point where tr(ad v | W ) = 0 because ad v is nilpotent. Here, we denoted by ad and Ad the adjoint representations of g and G respectively; recall that Ad exp(v) = e adv . This implies that ϕ preserves Haar measures of W and thus µ(pE) = ψ d (π W (pE)) = ψ d (p W ϕ(π W (E)) = ψ d (π W (E)) = µ(E).
We conclude that µ is a Haar measure on P, so the first part of the statement is proved.
It is worth observing that the area factor implicitly depends on the fixed group W. We are now ready to prove our first main result. (0, ∞). We define the measure µ, supported on Σ, by
We conclude that, by the definition of the area factor in Lemma 3.2, where the last equality follows from Lemma 3.1.
We conclude this section with some applications of Theorem 1.1. We start by proving the first part in the statement of Corollary 1.2 about the relation between Hausdorff and spherical Hausdorff measures on rectifiable sets; the second part of Corollary 1.2, concerning the same application in the setting of the Heisenberg group endowed with a rotationally invariant distance, will be proved in Proposition 5.4 Proof of Corollary 1.2, first part. If P ∈ T G,G ′ , let a(P) as the constant such that (18) S Q−m P = a(P)H Q−m P, Since Σ is arbitrary, we can apply this equality to Σ = P ∈ W W to see that . (2) are now clear. with the isodiametric problem on W about maximizing the measure of subsets of W with diameter at most 1 (see [42]). This task is a very demanding one already in the Heisenberg group endowed with the Carnot-Carathéodory distance, see [27].

Continuity of a and
We now prove a statement about weak* convergence of measures of level sets of C 1 H functions; this will be used in the subsequent Corollary 3.6 as well as later in the proof of the coarea formula. We note that the proof of Lemma 3.4 relies on the Area formula (1): we are not aware of any alternative strategy. H (Ω; G ′ ) and a point o ∈ Ω that is split-regular for g. Let m denote the homogeneous dimension of G ′ and, for b ∈ G ′ and λ > 0, define Then, the weak* convergence of measures holds. Moreover, the convergence is uniform with respect to b ∈ G ′ , i.e., for every χ ∈ C c (G) and every ε > 0 there isλ > 0 such that Proof. Up to replacing g with the function x → g(o) −1 g(ox), we can assume o = 0 and g(o) = 0; in particular, Σ λ,b = δ 1/λ ({p ∈ Ω : g(p) = δ λ b}). Notice that, by Lemma 2.10, Σ λ,b = ∅ for all b in a neighborhood of 0 and λ small enough.
Possibly restricting Ω, we can assume that there exists a splitting G = W · V, open sets A ⊂ W, B ⊂ G ′ and a map ϕ : A × B → V such that the statements of Lemma 2.10 hold. If p ∈ Σ λ,b , then there is a ∈ A such that p = δ 1/λ (aϕ(a, δ λ b)) = δ 1/λ aϕ λ (δ 1/λ a, b), where ϕ λ (a, b) := δ 1/λ ϕ(δ λ a, δ λ b). In particular, Σ λ,b is the intrinsic graph of ϕ λ (·, b) : Denoting by ϕ 0 : W × G ′ → V the implicit function associated with D H g(0), we have by Lemma 2.12 that ϕ λ → ϕ 0 uniformly on compact subsets of W × G ′ . Moreover where the convergence is in the topology of W V and it is uniform when (a, b) belong to a compact set of W × G ′ . Therefore, using the area formula of Theorem 1.1, for every χ ∈ C c (G) we have where the limit is uniform when b belongs to a compact subset of G ′ . Let us show that the convergence is actually uniform on G ′ .
Since g is Lipschitz continuous in a neighborhood of 0, there is a positive constant C such that ρ ′ (0, g(δ λ p)) ≤ Cλ for all p ∈ spt χ and λ small enough. Therefore, if ρ ′ (0, b) > C, then spt χ ∩ Σ λ,b = ∅. Possibly increasing C, we can assume that spt χ ∩ {D H g(o) = b} = ∅ for all b such that ρ ′ (0, b) > C. Therefore, the uniformity of the limit (19) for b ∈ B G ′ (0, C) implies uniformity for all b ∈ G ′ . This completes the proof.
In the proof of the following corollary, we will need this simple lemma: Lemma 3.5. Let θ be a Haar measure and ρ a homogeneous distance on a homogeneous group P. Then θ(∂ U ρ (0, R)) = 0 for all R > 0.
Proof. By homogeneity, there holds Corollary 3.6. There exists a continuous function d : T G,G ′ → (0, +∞) with the following property. If R ⊂ G is a (G; G ′ )-rectifiable set and Q, m denote the homogeneous dimensions of G, G ′ , respectively, then Moreover, if R is a C 1 H submanifold, then the equality in (20) holds at every p ∈ R. Clearly, d depends on whether the measure ψ Q−m under consideration is the Hausdorff or the spherical one.
We conclude this section with the following result, similar in spirit to Lemma 3.4. It will be used in the proof of Lemma 4.6.
Corollary 3.7. Suppose that, for n ∈ N, L n : G → G ′ is a homogeneous morphism and that the L n converge to a surjective homogeneous morphism L : G → G ′ such that ker L splits G. Then the following weak* convergence of measures holds: where Q is the homogeneous dimension of G and m is the homogeneous dimension of G ′ . More precisely, given a function χ ∈ C c (G) and ε > 0, there exists N ∈ N such that for all n ≥ N and s ∈ G ′ Proof. Denote by W := ker L and let G = W · V a splitting. Recall that V and G ′ are also vector spaces, the morphisms L n are linear maps and that L| V : V → G ′ is an isomorphism. Therefore, there exists N ∈ N such that L n | V is an isomorphism for all n ≥ N . For all such n and s ∈ G ′ , define φ s n : W → V by φ s n (w) := L n | −1 V (L n (w) −1 s). Notice that {L n = s} is the intrinsic graph of φ s n . Let φ s ∞ : W → V the function whose intrinsic graph is {L = s}: it is clear that φ s n (w) → φ s ∞ (w) uniformly on compact sets in the variables (w, s) ∈ W × G ′ .
Fix χ ∈ C c (G). Then where the functionsχ n : (s, w) → χ(wφ s n (w)) are continuous and uniformly converge to (s, w) → (wL| −1 V (s)) as n → ∞. Moreover, A(ker L n ) → 1. This completes the proof. Our aim is to prove Theorem 1.3, which by Proposition 2.2 will be a consequence of the following Theorem 4.1: here, C(P, L) denotes the coarea factor corresponding to a homogeneous subgroup P of G and a homogeneous morphism L : G → L; the coarea factor is going to be defined later in Proposition 4.5. The function C(P, L) is continuous in P and L, see Lemma 4.6. and assume that all points in Ω are split-regular for f , so that Σ := {p ∈ Ω : f (p) = 0} is a C 1 H submanifold. Consider a function u : Ω → L such that uf ∈ C 1 H (Ω; K) and assume that for ψ Q−m -a.e. p ∈ Σ, either D H (uf ) p | T H p Σ is not surjective on L, or p is split-regular for uf .
defined on Borel sets, is a locally finite measure; (iii) the Radon-Nikodym density Θ of µ Σ,u with respect to ψ Q−m Σ of is locally bounded and Remark 4.2. Let us prove that the differential D H (uf ) p | T H p Σ depends only on the restriction of u to Σ and, moreover, that it does not depend on the choice of the defining function f for Σ. In particular, in view of Proposition 4.5 also the coarea factor C(T H p Σ, D H (uf ) p ) depends only on the restriction of u to Σ.
Let v ∈ T H p Σ; then there exist sequences r j → 0 + and q j → p such that q j ∈ Σ and v = lim j→∞ δ 1/rj (p −1 q j ). In particular, q −1 j pδ rj v ρ = o(r j ) and, by Lemma 2.5, lim This proves what claimed.
The proof of Theorem 4.1 is divided into several steps. We start by proving that µ Σ,u is a well defined locally finite measure concentrated on Σ; this uses an abstract coarea inequality. Then we consider the linear case in order to apply a blow-up argument; in doing so, we will define the coarea factor. We finally consider separately "good points", i.e., those where D H (uf )| T H Σ has full rank, and "bad points", where D H (uf )| T H Σ is not surjective: at good points the blow-up argument applies, while the set of bad points is negligible by an argument similar to the proof of the coarea inequality.

Coarea Inequality.
In this section we prove Proposition 4.4, which is a consequence of the following Lemma 4.3; the latter is basically [13,Theorem 2.10.25], with a slightly different use of the Lipschitz constant. See also [28,Theorem 1.4] and [11,Lemma 3.5].
where H β is the β-dimensional Hausdorff measure on (Y, d Y ). Let u : X → Y be a locally Lipschitz function and for ε > 0 consider Lip ε (u).
Then, for every α ≥ β and every Borel set

Moreover, the set function
The proof is standard. In our setting, the "abstract" coarea inequality translates as follows. (iii) for every compact K ⊂ Σ, the coarea inequality holds for a suitable C = C(L) > 0; (iv) µ Σ,u is a Borel measure on Ω, µ Σ,u ≪ ψ Q−m Σ with locally bounded density.
Proof. The local Lipschitz continuity of u| Σ follows from Lemma 2.5 because of the assumption uf ∈ C 1 H (Ω; K) and the fact that u| Σ = uf | Σ . As already noticed in the proof of Lemma 4.3, statement (ii) follows from [13, 2.10.26]; the careful reader will observe that [13, 2.10.26] is stated only when ψ ℓ = H ℓ , but its proof easily adapts to the case ψ ℓ = S ℓ . Concerning statement (iii), we notice that ψ Q−m (K) < ∞ because the measure ψ Q−m Σ is locally finite by Lemma 2.14 and the area formula (Theorem 1.1): in particular, one can apply Lemma 4.3. Statement (iv) is now a consequence of statement (iii) and the Radon-Nikodym Theorem, which can be applied because ψ Q−m Σ is doubling by (12) (see, e.g., [43]).

4.3.
Linear case: definition of the coarea factor. In the following Proposition 4.5 we prove the coarea formula in a "linear" case, and in doing so we will introduce the coarea factor. We are going to consider a homogeneous subgroup P of G that is also a C 1 H submanifold. We observe that this implies that P coincides with its homogeneous tangent subgroup; in particular, P is normal and it is the kernel of a surjective homogeneous morphism on G. Then, µ P,L is either null or a Haar measure on P. In particular, there exists C(P, L) ≥ 0, which we call coarea factor, such that (23) µ P,L = C(P, L) ψ Q−m P. Proof. Since L is Lipschitz on P, we can apply Proposition 4.4 and obtain that µ P,L is a well defined Borel regular measure that is also absolutely continuous with respect to ψ Q−m P and finite on bounded sets. If L(P) = L, then µ P,L = 0 and thus (23) holds with C(P, L) = 0.
If L(P) = L, then we will show that µ P,L is a Haar measure on P, which is equivalent to (23) with C(P, L) > 0. For s ∈ L let P s := L −1 (s). Since P s is a coset of P 0 , ψ Q−m−ℓ P s is the push-forward of ψ Q−m−ℓ P 0 (which is a Haar measure on P 0 ) via a left translation. It follows that µ P,L is nonzero on nonempty open subsets of P. We need only to show that µ P,L is left-invariant: let p ∈ P and choose a Borel set A ⊂ P. For every s ∈ L we have p −1 P s = {q ∈ P : L(pq) = s} = P L(p) −1 s . By left invariance of ψ Q−m−ℓ and ψ ℓ , we have as wished.
We now prove a continuity property for the coarea factor C(P, L). We agree that, when L : G → L is defined on the whole G, the symbol C(P, L) stands for C(P, L| P ).
Lemma 4.6. Assume that, for n ∈ N, surjective homogeneous morphisms F, F n : G → M and homogeneous maps L, L n : G → L are given in such a way that (i) LF and L n F n are homogeneous morphisms G → K; (ii) ker F and ker(LF ) split G; (iii) F n → F and L n → L on G as n → ∞.
Then C(ker F n , L n ) → C(ker F, L) as n → ∞.
Proof. Set P n := ker F n and P := ker F ; let V be a complementary subgroup to P. Then, for large enough n, P n · V is a splitting of G and the subgroup P n is the intrinsic graph P → V of a homogeneous map φ n ∈ C 1 P,V (P). Observe that φ n → 0 locally uniformly on P because P n → P. This, together with Lemma 3.2 and the continuity of the area factor by Lemma 3.2, implies that ψ Q−m P n converges weakly* to ψ Q−m P. Therefore, by Proposition 4.5 we have only to show that (24) µ Pn,Ln * ⇀ µ P,L .
If L| P is surjective, also LF is surjective. Since ker(LF ) splits G, then (24) follows from Corollary 3.7. If L| P is not surjective, we can without loss of generality suppose that L n | Pn is surjective for all n. By homogeneity, it suffices to prove that µ Pn,Ln (B G (0, 1)) → 0. We have where the last inequality holds because, considering p ∈ P n such that L n (p) = s −1 , we have by Lemma 3.1 Thus, we have to prove that ψ ℓ (L n (B G (0, 1) ∩ P n )) → 0; notice that L n (B G (0, 1) ∩ P n ) converges in the Hausdorff distance to L(B G (0, 1) ∩ P), which is a compact set contained in a strict subspace of L. As ψ ℓ is a Haar measure on L, we have ψ ℓ (L n (B G (0, 1) ∩ P n ) → 0 as n → ∞.

Good Points.
By "good" point o ∈ Σ we mean a point where the differential D H (uf )| T H o Σ is surjective onto L; the following Proposition 4.7 shows that the Radon-Nikodym density Θ of µ Σ,u with respect to ψ Q−m Σ can be explicitly computed at its Lebesgue points and coincides with the coarea factor. Notice that almost every o ∈ Σ is a Lebesgue point for Θ, in the sense that (25) lim Proposition 4.7. Under the assumptions and notation of Theorem 4.1, one has that the equality .
We are going to prove (26) for all o ∈ Σ such that D H (uf )| T H 0 Σ is onto L, o is split-regular for uf and (25) holds; up to left translations, we may assume that o = 0 and u(0) = 0. For every Borel set A ⊂ G and λ > 0 we have, on the one hand On the other hand, where u λ (p) := δ 1/λ u(δ λ p). Therefore, one has the equality of measures We now compute the weak* limits as λ → 0 + of each side of (27). Concerning the left-hand side, for every χ ∈ C c (G) one has Let r > 0 be such that spt χ ⊂ U(0, r), then for a suitable positive C. Exploiting (25) one gets the last equality following from Lemma 3.4. We now consider the right-hand side of (27); setting (uf ) λ (p) := δ 1/λ ((uf )(δ λ p)), for every χ ∈ C c (G) one has where we used Lemma 3.4. The definition of coarea factor then gives The statement is now a consequence of (27), (28) and (29). 4.5. Bad points. In contrast with "good" ones, "bad" points are those points p where (D H (uf ))(p)| T H p Σ is not surjective. The following lemma states that they are µ Σ,u -negligible: a posteriori, this is consistent with the fact that, by definition, the coarea factor is null at such points.
It is enough to show that µ Σ,u (E) = 0 for an arbitrary compact subset E of {p ∈ Σ : D H (uf )(p)| T H p Σ is not onto L}, which is closed. We have ψ Q−m (E) < ∞. Fix ε > 0; by the compactness of E and the locally uniform differentiability of both f and uf , there exists r > 0 such that B(E, r) ⊂ Ω and, for all p ∈ E and all q ∈ Σ ∩ U(p, r), the inequalities dist(q, pT H p Σ) ≤ ερ G (p, q), r) ). Fixing a positive integer j > 1/r, one can cover E by countably many closed sets {B j i } i of diameter d j i := diam B j i belonging to the class E and such that (30) d j i < 1/j, for all i, and Imitating the proof of [11, Lemma 3.5], we define the functions g j i : L → [0, 1] by . Note that, using the standard notation ψ Q−m−ℓ δ for the pre-measures used in the Carathéodory construction, one has for all y ∈ Y . Then one gets, using upper integrals, where the inequality marked by * follow from Fatou's Lemma. We claim that for a suitable C(ε, L) > 0 such that lim ε→0 + C(ε, L) = 0.
Let us prove (33). Fix some B = B j i ; we can assume that B intersects E in at least a point p, which implies in particular that B ⊂ B(E, 1/j). Without loss of generality, suppose that p = 0 and (uf )(p) = 0; we know that for every q ∈ B ∩ Σ dist(q, T H 0 Σ) ≤ ε q G and ρ K (u(q), D H (uf ) 0 (q)) ≤ M ε q G .
Observing that D H (uf ) 0 has Lipschitz constant at most M , we get Denoting by L ′ the homogeneous subgroup D H (uf ) 0 (T H 0 Σ), which is strictly contained in L, and using the fact that u(B ∩ Σ) ⊂ L, we conclude that where we also used the fact that the Lipschitz constant of u| B∩Σ = (uf )| B∩Σ is at most M . By homogeneity one has The claim (33) follows on letting where the supremum is taken among proper homogeneous subgroups of L. The fact that lim ε→0 + C(ε, L) = 0 can be easily checked in linear coordinates on the vector space L, by comparing ρ L with the Euclidean distance and noting that ψ ℓ is a multiple of the Lebesgue measure.
Combining (33), (32) and (30), we obtain and, letting j → ∞, we deduce by Fatou's Lemma that The proof is accomplished by letting ε → 0 + .  A direct consequence is Corollary 1.4, where we assume that K = L × M is a direct product: Proof of Corollary 1.4. It is enough to prove the statement in case R is a C 1 H submanifold; actually, we can also assume that there exists f ∈ C 1 H (Ω; M) such that R = Σ := {p ∈ Ω : f (p) = 0} and all points in Ω are split-regular for f . Since In particular, condition (5) now implies (3), and the statement directly follows from Theorem 1.3.

Heisenberg groups
The most notable examples of Carnot groups are provided by Heisenberg groups. For an integer n ≥ 1, the n-th Heisenberg group H n is the stratified Lie group associated with the step 2 algebra V = V 1 ⊕ V 2 defined by V 1 = span{X 1 , . . . , X n , Y 1 , . . . , Y n }, [X i , Y j ] = δ ij T for every i, j = 1, . . . , n.
5.1. Area formula in Heisenberg groups. We provide an explicit representation for the spherical measure on vertical subgroups of H n : Proposition 5.1. Assume that H n is endowed with a rotationally invariant homogeneous distance and let 1 ≤ k ≤ n. Then, there exists a constant c(n, k) such that for every vertical subgroup P ∈ T H n ,R k c(n, k)S 2n+2−k P = H 2n+1−k E P, where H 2n+1−k E denotes the Euclidean Hausdorff measure on R 2n+1 ≡ H n .
Proof. Let P ∈ T H n ,R k be a fixed vertical subgroup; by [16,Lemma 3.26] there exists a complementary Abelian horizontal subgroup V = V × {0}, for a proper k-dimensional subspace V ⊂ V 1 . Let W be a (2n − k)-dimensional complementary subspace of V in V 1 and set W := W × V 2 , which is a vertical subgroup that is complementary to V. Let P ⊂ V 1 such that P = P × V 2 .
The area formula of [7, Theorem 1.2], together with [7, Theorem 2.12 and Proposition 2.13] from the same paper, provide a constant c(n, k) > 0 such that (35) c(n, k)S 2n+2−k P = Φ # (J φ φ H 2n+1−k E W), 3 The terminology "rotationally invariant" might be misleading in H n for n > 1, as not all rotations around the T axis are isometries where J φ φ is the intrinsic Jacobian of φ as in [7, Definition 2.14] and Φ is the intrinsic graph map. On the other side, the Euclidean area formula gives where F : W → P is defined by F (x, y, t) := (f (x, y), t) for every (x, y) ∈ W and JF is the Euclidean area factor. As a matter of fact, using the equality f = φ, one has J φ φ = JF and the statement immediately follows from (35) and (36).
Remark 5.2. Proposition 5.1 holds, with no changes in the proof, in the more general case H n is endowed with a homogeneous distance that is (2n + 1 − k)vertically symmetric according to [7,Definition 2.19].
Remark 5.3. When H n is endowed with a rotationally invariant distance ρ, then for every pair (P, P ′ ) of one-codimensional homogeneous subgroups of H n , there exist an isometry (H n , ρ) → (H n , ρ) that maps P to P ′ . The proof is left to the reader.
The following proposition completes the proof of Corollary 1.2.
Proposition 5.4. If H n is endowed with a rotationally invariant homogeneous distance and G ′ = R, then the function a in Corollary 1.2 is constant, i.e., there exists C ∈ [1, 2 2n+1 ] such that Proof. When G = H n and G ′ = R, then the function a defined in (18)  Similarly, Corollary 3.6 can be improved when G is the Heisenberg group endowed with a rotationally invariant distance.
Corollary 5.5. Assume G is the Heisenberg group H n endowed with a rotationally invariant distance and G ′ = R m for some 1 ≤ m ≤ n; if ψ 2n+2−m is the spherical Hausdorff measure, then the function d in Corollary 3.6 is constant.
If m = 1 and ψ 2n+2−m is the Hausdorff measure, then the function d in Corollary 3.6 is constant.
Proof. Concerning the first part of the statement, let W ∈ T H n ,R m be fixed; by Proposition 5.1 we have d(W) = lim r→0 + S 2n+2−m (W ∩ U(0, r)) r 2n+2−m = S 2n+2−m (W ∩ U(0, 1)) = c(n, m)H 2n+1−m E (W ∩ U(0, 1)) and the latter quantity does not depend on W by rotational invariance of the distance. The second part of the statement is an immediate consequence of Remark 5.3.

5.2.
Coarea formula in Heisenberg groups. When one considers spherical measures in the Heisenberg group endowed with a rotationally invariant distance, then the coarea factor coincides up to a multiplicative constant with the quantity J R u(p) := det(L • L T ) 1/2 , L := D H u p | T H p R . We prove this fact. Proposition 5.6. Consider the Heisenberg group H n endowed with a rotationally invariant distance. Let P ∈ T H n ,R m be a vertical subgroup of topological dimension 2n + 1 − m and let L : P → R ℓ be a homogeneous morphism; assume 1 ≤ m + ℓ ≤ n. Proof. If L is not onto R ℓ , then the statement is true. We assume that L is surjective. By Proposition 5. We now have all the tools needed in order to prove our coarea formula in Heisenberg groups.
Proof of Theorem 1.7. The first part of the statement is an immediate consequence of Corollary 1.4 and the fact that, if D H u p | T H p R is surjective on R ℓ , then T H p R ∩ ker D H u p is a vertical subgroup of dimension 2n + 1 − m − ℓ ≥ n + 1, and by [16,Lemma 3.26] it admits a complementary (horizontal) subgroup.
The second part of the statement is now a consequence of Proposition 5.6; clearly, one has c = c(n, m + ℓ)/c(n, m) according to the constants introduced in Proposition 5.1.