ON THE STRUCTURE OF FINITE PERIMETER SETS IN STEP 2 CARNOT GROUPS

BRUNO FRANCHI, RAUL SERAPIONI, AND FRANCESCO SERRA CASSANO

1. Introduction

This paper is intended to contribute to the general program of developing Geometric Measure Theory in Carnot groups. A general account of the many facets and contributions to this research stream is far beyond the aim of this introductions. Results strictly related to ours are quoted in the body of the paper and here we restrict ourselves to recommend the reader to the general monographies [18], [43], [42], [41], [64], [66], [53], to the papers [4], [5], [6], [12], [15], [16], [29], [32], [37], [47], [57], [56], [67] and to the references therein.

One of the sources of the current interest in Carnot groups and their geometry is that they appear as tangent groups of subriemannian manifolds (i.e. differentiable manifolds endowed with a distribution of k-planes and a fiber inner product on the distribution). Subriemannian structures arise for instance when dealing with non-holonomic systems in mechanics and in control theory (see e.g. the recent survey in [53], together with its rich bibliography). Carnot groups are not Euclidean at any scale but, like Euclidean spaces, have a rich structure of dilations and translations. In a sense they play versus subriemannian manifolds the same role played by Euclidean spaces versus Riemannian manifolds.

All the results of the present paper deal with the notion of intrinsically rectifiable – or grouprectifiable – sets (of codimension 1) inside a Carnot group. They can be summarized in the following three points.

• We prove that, in a step 2 Carnot group, the measure theoretic boundary (or the reduced boundary) of sets with finite intrinsic perimeter is group-rectifiable. We recall that a set E has finite perimeter when its characteristic function 1_{E} has bounded variation, and the rectifiability result gives geometric substance to such an abstract functional definition. This result extends De Giorgi’s classical structure theorem in Euclidean spaces ([19],[20]), (as well as our previous theorem in Heisenberg groups ([32])). Among the consequences we explicitly mention a general divergence formula in step 2 Carnot groups. These results have been previously announced in [34].

• A second group of results deals with the comparisons between the intrinsic notion of group rectifiability and the Euclidean one – both notions being precisely defined below. We observe that comparing the different features of the geometries induced, on the same ℝ^{n}, by different subriemannian structures is a task up to now largely incomplete. Here we show that codimension 1 Euclidean rectifiable sets are also codimension 1 group rectifiable, whereas the opposite is false. Even if this statement is not surprising, its proof is far from being trivial. Notice that the similar statement relating Euclidean C^{1} and intrinsically C^{1} codimension 1 submanifolds is false: see [32], Remark 6.2. The same results in the special case of the Heisenberg group have been proved recently in [8].

Date: May 24, 2002.

The authors are supported by GNAMPA of INdAM, project ”Analysis in metric spaces and subelliptic equations”.

B.F. is supported by University of Bologna, Italy, funds for selected research topics.

R.S. is supported by MURST, Italy, and University of Trento, Italy.

F.S.C. is supported by MURST, Italy, and University of Trento, Italy.

• Finally we point the attention to one feature in which the geometry of Carnot groups of step larger than 2 is definitely different from that of step 2 groups. To explain better this point we recall the following Lemma, intuitively true in Euclidean spaces,

Lemma: If C is a sufficiently regular, invariant by dilations open subset of ℝ^{n} – i.e. C is a cone with vertex at the origin – and if the normal vector field ν to ∂C is a constant vector field then the cone C is actually a halfspace.

This Lemma lies at the centre of De Giorgi’s approach to the rectifiability: indeed eventually it yields the existence of an approximate tangent plane at each point of the reduced boundary of a finite perimeter set. Recall that the existence of approximate tangent planes together with density estimates is at the core of most proofs of rectifiability. Notice now that the statement of the Lemma makes sense also in Carnot groups: Euclidean dilations are substituted by the non homogeneous dilations of the group, while the horizontal normal (see Definition 2.18) to ∂C takes the place of the Euclidean norm. We prove here that the Lemma is true in step 2 Carnot groups – and this allows us to prove De Giorgi’s theorem in the same context – while, on the contrary, the Lemma is false for Carnot groups of step larger than 2! Indeed we provide an example of a non flat intrinsic cone (i.e. a set that is invariant under group dilations) in the step 3 Engel’s group, with constant horizontal normal.

Going back to the first point, we give now a less vague description of the rectifiability result to show a feature that we find worthwhile of attention i.e. that the statement of rectifiabilty theorem in Carnot groups is essentially the same as it is in Euclidean spaces, provided that all notions involved in the Euclidean statement be appropriately redefined.

A Carnot group 𝔾 of step 2 is a connected, simply connected Lie group whose Lie algebra 𝔤 admits a step 2 stratification, i.e. there exist linear subspaces V_{1}, V_{2} such that

𝔤 = V_{1} ⊕ V_{2}, [V_{1}, V_{i}] = V_{i+1}, V_{2} ≠ {0}, V_{i} = {0} if i > 2,

where [V_{1}, V_{i}] is the subspace of 𝔤 generated by the commutators [X, Y] with X ∈ V_{1} and Y ∈ V_{i}. Through the exponential map, 𝔾 can be identified with ℝ^{n} endowed with a non commutative group operation and a non isotropic family of dilations (see Section 2). With the first layer V_{1} of the stratification, we can canonically associate a subbundle of the tangent bundle of 𝔾, called the horizontal bundle H𝔾. The Carnot-Carathéodory distance d_{c} on 𝔾 is defined as usual through subunit (or admissible) curves, i.e. continuously differentiable curves whose tangent vectors are sections of H𝔾.

This metric d_{c} yields natural definitions of Hausdorff measures and of metric dimensions of 𝔾 and of its subsets. We stress here that the metric dimension of 𝔾 is Q := dim V_{1} + 2dim V_{2} > n.

As for the differentiable structure in 𝔾, any basis X_{1}, …, X_{m_{1}} of the horizontal layer V_{1} of 𝔤 takes the role of the first order partial derivatives ∂_{1}, …, ∂_{n} in ℝ^{n}. As a consequence, there are intrinsic notions of gradient ∇_{𝔾} of real valued functions, of divergence div_{𝔾} of horizontal vector fields and finally of Bounded Variation functions (see Subsection 2.2).

A set E ⊂ ℝ^{n} is a locally finite 𝔾-perimeter set if its characteristic function 1_{E} is in BV_{loc}(𝔾), that is the space of intrinsic locally BV functions in 𝔾, and its perimeter measure |∂E|_{𝔾} is the total variation of the measure ∇_{𝔾}1_{E} (see Subsection 2.3).

We remind now that De Giorgi’s celebrated structure theorem in Euclidean spaces states that if E ⊂ ℝ^{n} is a set of locally finite perimeter, then the associated perimeter measure |∂E| is concentrated on a portion of the topological boundary ∂E, the so-called reduced boundary ∂^{∗}E ⊂ ∂E. In addition, ∂^{∗}E is H^{d−1}-rectifiable, i.e. ∂^{∗}E, up to a set of (d − 1)-Hausdorff measure zero, is a countable union of compact subsets of C^{1} submanifolds and the perimeter measure is the (n − 1)-Hausdorff measure of the reduced boundary.

If in the spirit of De Giorgi’s theorem we want to describe the structure of sets of finite intrinsic perimeter in a Carnot group 𝔾, we need a natural notion of rectifiable subsets, and in this perspective, the crucial missing point is – as already pointed in [32] and in [33] – the correct definition of intrinsic C^{1}-regular submanifold of 𝔾. In fact, there is a classical notion of rectifiability in a metric space that goes back to Federer (see [26] 3.2.14) that has been recently used by Ambrosio & Kirchheim (see [4], [5]) in the contest of a theory of currents in metric spaces (as for the rectifiability in metric spaces see for instance [51], [46], [52], [36], [62] and also the monografy [50] and the references therein). According to this notion, a regular hypersurface in a metric space should be the image of an open subset of an Euclidean space via a Lipschitz map. Unfortunately, such a notion does not fit the geometry of Carnot groups, that indeed would be, according with this definition, purely unrectifiable (see [4]). Hence we found convenient to define C_{𝔾}^{1} manifolds as non-critical level sets of C_{𝔾}^{1} real valued functions (see subsection 2.5) and in addition the normalized horizontal gradient of the defining function plays the role of a normal to the surface, that is, notice, an horizontal section (see Theorem 2.35). As proved in [33] these objects have the correct Q − 1 intrinsic metric dimension.

Keeping in mind the notions just illustrated, the statement of our structure theorem reads precisely as De Giorgi’s theorem in the Euclidean setting provided all the objects involved are replaced by their intrinsic counterparts.

In general, throughout the paper, we keep our statements as much general as possible. This means that many statements that hold in general Carnot groups are proved in this setting, even if not necessary for the final result. This has been done to focus more the attention on the key points where the step 2 group structure is crucial.

We have to adknowledge – and it is a pleasure to do it – that in a friendly and informal talk with a much younger second author, several years ago, when the analysis in Carnot spaces was less fashionable than it is now, De Giorgi already raised the problem of studying rectifiability (and possibly regularity of minimal surfaces) in subriemannian manifolds.

It is also a pleasure to thanks several friends: Martin Reimann that inspired us the counterexample in the Engel’s group and generously shared with us his ideas, together with Roberto Monti that found a preliminary form of the counterexample itself. We thank also Luigi Ambrosio for his unvaluable constant support, and Pertti Mattila for his hospitality and for the many hours he spent talking with the third author.

Let us give now a short sketch of the structure of the present paper.

In section 2 we give the basic definition and we collect more or less know results on Carnot groups, bounded variation functions and regular hypersurfaces in groups. Moreover, a Whitney type extension theorem in Carnot groups is stated. Its proof is postponed to the Appendix. In Section 3 we prove our main result about the structure of the boundary of finite perimeter sets and, as a consequence, a generalized divergence theorem. In addition, an example showing the impossibility of extending De Giorgi’s blow-up technique to groups of step 3 is provided. In Section 4, we give an estimate of the Hausdorff intrinsic dimension of the set of characteristic points of a continuously differentiable hypersurface. As a consequence, we prove that codimension 1 Euclidean rectifiable sets are also codimension 1 group rectifiable.

2. Notations and Preliminary Results

2.1. Carnot Groups. We recall some definitions and a few previous results. We state these results in a setting more general than needed for our main theorem. Indeed this section will concern Carnot groups of any step k (see below for a precise definition) even if our main result holds in step 2 groups.

A Carnot group 𝔾 of step k (see [27], [44], [55], [45], [56], [65], and [66]) is a connected, simply connected Lie group whose Lie algebra 𝔤 admits a step k stratification, i.e. there exist linear subspaces V_{1}, ..., V_{k} such that

𝔤 = V_{1} ⊕ ... ⊕ V_{k}, [V_{1}, V_{i}] = V_{i+1}, V_{k} ≠ {0}, V_{i} = {0} if i > k, (1)

where [V_{1}, V_{i}] is the subspace of 𝔤 generated by the commutators [X, Y] with X ∈ V_{1} and Y ∈ V_{i}. Let m_{i} = dim(V_{i}), for i = 1, …, k and h_{i} = m_{1} + ⋯ + m_{i} with h_{0} = 0 and, clearly, h_{k} = n. Choose a basis e_{1}, …, e_{n} of 𝔤 adapted to the stratification, that is such that

e_{h_{j−1}+1}, …, e_{h_{j}} is a base of V_{j} for each j = 1, …, k.

Let X = X_{1}, …, X_{n} be the family of left invariant vector fields such that X_{i}(0) = e_{i}. Given (1), the subset X_{1}, …, X_{m_{1}} generates by commutations all the other vector fields; we will refer to X_{1}, …, X_{m_{1}} as generating vector fields of the group. The exponential map is a one to one map from 𝔤 onto 𝔾, i.e. any p ∈ 𝔾 can be written in a unique way as p = exp(p_{1}X_{1} + ⋯ + p_{n}X_{n}). Using these exponential coordinates, we identify p with the n-tuple (p_{1}, …, p_{n}) ∈ ℝ^{n} and we identify 𝔾 with (ℝ^{n}, ·) where the explicit expression of the group operation · is determined by the Campbell-Hausdorff formula (see [27]) and some of its features are described in the following Proposition 2.1. If p ∈ 𝔾 and i = 1, …, k, we put p^{i} = (p_{h_{i−1}+1}, …, p_{h_{i}}) ∈ ℝ^{m_{i}}, so that we can also identify p with [p^{1}, …, p^{k}] ∈ ℝ^{m_{1}} × ⋯ × ℝ^{m_{k}} = ℝ^{n}.

The subbundle of the tangent bundle T 𝔾 that is spanned by the vector fields X_{1}, …, X_{m_{1}} plays a particularly important role in the theory, it is called the horizontal bundle H𝔾; the fibers of H𝔾 are

H𝔾_{x} = span {X_{1}(x), …, X_{m_{1}}(x)}, x ∈ 𝔾.

A subriemannian structure is defined on 𝔾, endowing each fiber of H𝔾 with a scalar product ⟨·, ·⟩_{x} and with a norm | · |_{x} that make the basis X_{1}(x), …, X_{m_{1}}(x) an orthonormal basis. That is if v = ∑_{i=1}^{m_{1}} v_{i}X_{i}(x) = (v_{1}, …, v_{m_{1}}) and w = ∑_{i=1}^{m_{1}} w_{i}X_{i}(x) = (w_{1}, …, w_{m_{1}}) are in H𝔾_{x}, then ⟨v, w⟩_{x} := ∑_{j=1}^{m_{1}} v_{j}w_{j} and |v|_{x}^{2} := ⟨v, v⟩_{x}. The sections of H𝔾 are called horizontal sections, a vector of H𝔾_{x} is an horizontal vector while any vector in T 𝔾_{x} that is not horizontal is a vertical vector. Each horizontal section is identified by its canonical coordinates with respect to this moving frame X_{1}(x), …, X_{m_{1}}(x). This way, an horizontal section ϕ is identified with a function ϕ = (ϕ_{1}, …, ϕ_{m_{1}}) : ℝ^{n} → ℝ^{m_{1}}. When dealing with two such sections ϕ and ψ whose argument is not explicitly written, we drop the index x in the scalar product writing ⟨ψ, ϕ⟩ for ⟨ψ(x), ϕ(x)⟩_{x}. The same convention is adopted for the norm.

Two important families of automorphism of 𝔾 are the so called intrinsic translations and the intrinsic dilations of 𝔾. For any x ∈ 𝔾, the (left) translation τ_{x} : 𝔾 → 𝔾 is defined as

z ↦ τ_{x}z := x · z.

For any λ > 0, the dilation δ_{λ} : 𝔾 → 𝔾, is defined as

δ_{λ}(x_{1}, ..., x_{n}) = (λ^{α_{1}}x_{1}, ..., λ^{α_{n}}x_{n}), (2)

where α_{i} ∈ ℕ is called homogeneity of the variable x_{i} in 𝔾 (see [28] Chapter 1) and is defined as

(3) α_{j} = i whenever h_{i−1} + 1 ≤ j ≤ h_{i},

hence 1 = α_{1} = ... = α_{m_{1}} < α_{m_{1}+1} = 2 ≤ ... ≤ α_{n} = k.

We collect in the following proposition some more or less elementary properties of the group operation and of the canonical vector fields .

Proposition 2.1. The group product has the form

x · y = x + y + Q(x, y), ∀x, y ∈ ℝ^{n} (4)

where Q = (Q_{1}, …, Q_{n}) : ℝ^{n} × ℝ^{n} → ℝ^{n} and each Q_{i} is a homogeneous polynomial of degree α_{i} with respect to the intrinsic dilations of 𝔾 defined in (2), that is

Q_{i}(δ_{λ}x, δ_{λ}y) = λ^{α_{i}}Q_{i}(x, y), ∀x, y ∈ 𝔾.

Moreover, again ∀x, y ∈ 𝔾

Q_{1}(x, y) = ... = Q_{m_{1}}(x, y) = 0,

Q_{j}(x, 0) = Q_{j}(0, y) = 0 and Q_{j}(x, x) = Q_{j}(x, −x) = 0, for m_{1} < j ≤ n; (5)

finally,if h_{i−1} < j ≤ h_{i} and 2 ≤ i, then

Q_{j}(x, y) = Q_{j}(x_{1}, …, x_{h_{i−1}}, y_{1}, …, y_{h_{i−1}}), . (6)

Proof. For the first part see [64], Chapter 12, Section 5. The last statement follows from the homogeneity of Q_{j}.

Note that from Proposition 2.1 it follows that

δ_{λ}x · δ_{λ}y = δ_{λ}(x · y)

and that the inverse x^{−1} of an element x = (x_{1}, …, x_{n}) ∈ (ℝ^{n}, ·) has the form

x^{−1} = (−x_{1}, …, −x_{n}),

(see [28], Proposition 2.1 and also [44]).

Proposition 2.2. The vector fields X_{j} have polynomial coefficients and if h_{l−1} < j ≤ h_{l}, 1 ≤ l ≤ k,

X_{j}(x) = ∂_{j} + ∑_{i>h_{l}}^{n} q_{i,j}(x)∂_{i}, (7)

where q_{i,j}(x) = \frac{∂Q_{i}}{∂y_{j}} (x, y)|_{y=0} so that if h_{l−1} < i ≤ h_{l} then q_{i,j}(x) = q_{i,j}(x_{1}, ..., x_{h_{l−1}}) and q_{i,j}(0) = 0.

Proof. Because X_{j} is a left invariant vector field

(X_{j}f)(x) = \frac{∂f(x · y)}{∂y_{j}}|_{y=0}

now remember the form of x · y as described in the previous proposition

= \frac{∂f}{∂x_{j}} (x) + ∑_{i>h_{l}} \frac{∂Q_{i}}{∂y_{j}} (x, y)|_{y=0} \frac{∂f}{∂x_{i}} (x)

= \frac{∂f}{∂x_{j}} (x) + ∑_{i>h_{l}} q_{i,j}(x)\frac{∂f}{∂x_{i}} (x).

To conclude observe that from (4) in each polynomial Q_{i} there is no term depending on a single variable, hence q_{i,j}(0) = \frac{∂Q_{i}}{∂y_{j}} (x, y)|_{x,y=0} = 0.

An absolutely continuous curve γ : [0, T] → 𝔾 is a sub-unit curve with respect to X_{1}, …, X_{m_{1}} if it is an horizontal curve, that is if there are real measurable functions c_{1}(s), …, c_{m_{1}}(s), s ∈ [0, T] such that

γ̇(s) = ∑_{j=1}^{m_{1}} c_{j}(s)X_{j}(γ(s)), for a.e. s ∈ [0, T],

and if

∑_{j} c_{j}^{2} ≤ 1.

Definition 2.3. (Carnot-Carathéodory distance) If p, q ∈ 𝔾, their cc-distance d_{c}(p, q) is

d_{c}(p, q) = inf {T > 0 : there is a subunit curve γ with γ(0) = p, γ(T) = q} .

The set of subunit curves joining p and q is not empty, by Chow’s theorem, since by (1), the rank of the Lie algebra generated by X_{1}, …, X_{m_{1}} is n; hence d_{c} is a distance on 𝔾 inducing the same topology as the standard Euclidean distance. We shall denote U_{c}(p, r) and B_{c}(p, r) respectively the open and closed balls associated with d_{c}.

Several distances equivalent to d_{c} have been used in the literature. Later on, we shall use the following one, that can also be computed explicitly

d_{∞}(x, y) = d_{∞}(y^{−1} · x, 0),

where, if p = (p̃_{1}, …, p̃_{k}) ∈ ℝ^{m_{1}} × ⋯ × ℝ^{m_{k}} = ℝ^{n}, then

d_{∞}(p, 0) = max{ε_{j}||p^{j}||_{ℝ^{m_{j}}}^{1/j} , j = 1, …, k}. (8)

Here ε_{1} = 1, and ε_{2}, … ε_{k} ∈ (0, 1) are suitable positive constants depending on the group structure (see Theorem 5.1). As above, we shall denote U_{∞}(p, r) and B_{∞}(p, r) respectively the open and closed balls associated with d_{∞}. Both the cc-metric d_{c} and the metric d_{∞} are well behaved with respect to left translations and dilations, that is

\begin{matrix}d_{c}(z · x, z · y) = d_{c}(x, y) & , d_{c}(δ_{λ}(x), δ_{λ}(y)) = λd_{c}(x, y) \\ d_{∞}(z · x, z · y) = d_{∞}(x, y) & , d_{∞}(δ_{λ}(x), δ_{λ}(y)) = λd_{∞}(x, y) \end{matrix} (9)

for x, y, z ∈ 𝔾 and λ > 0.

Related with these distances, different Hausdorff measures, obtained by Carathédory’s construction as in [26] Section 2.10.2, are used in this paper: we denote by H^{m} the m-dimensional Hausdorff measure obtained from the Euclidean distance in ℝ^{n} ≃ 𝔾, by H_{c}^{m} the m-dimensional Hausdorff measure obtained from the distance d_{c} in 𝔾, and by H_{∞}^{m} the m-dimensional Hausdorff measure obtained from the distance d_{∞} in 𝔾. Analogously, S^{m}, S_{c}^{m}, and S_{∞}^{m} denote the corresponding spherical Hausdorff measures.

The integer

Q = ∑_{j=1}^{n} α_{j} = ∑_{i=1}^{k} i dim V_{i} (10)

is the homogeneous dimension of 𝔾. It is also the Hausdorff dimension of ℝ^{n} with respect to the cc-distance d_{c} (see [54]).

The n-dimensional Lebesgue measure L^{n}, is the Haar measure of the group 𝔾. Hence if E ⊂ ℝ^{n} is measurable, then L^{n}(x · E) = L^{n}(E) for all x ∈ 𝔾. Moreover, if λ > 0 then L^{n}(δ_{λ}(E)) = λ^{Q}L^{n}(E). We explicitly observe that

L^{n}(U_{c}(p, r)) = r^{Q}L^{n}(U_{c}(p, 1)) = r^{Q}L^{n}(U_{c}(0, 1)). (11)

All the spaces L^{p}(𝔾) that appear throughout this paper are defined with respect to the L^{n} Lebesgue measure.

The following regularity properties hold not only for d_{c} and d_{∞}, but for all distances in 𝔾 that are translation invariant and homogeneous of degree 1 with respect to the natural dilations of 𝔾 in the sense of (9).

Proposition 2.4. Let d be a distance in 𝔾 such that

d(z · x, z · y) = d(x, y) , d(δ_{λ}(x), δ_{λ}(y)) = λd(x, y)

for x, y, z ∈ 𝔾 and λ > 0, and denote by B_{d} the closed d-balls. Then

diam_{d} (B_{d}(x, r)) = 2r, for r > 0; (i)

moreover, if μ is an s-homogeneous Radon measure for some s > 0, then

μ(∂B_{d}(0, r)) = 0, for r > 0. (ii)

Proof. To prove (i), we assume without loss of generality x = 0 and r = 1. Obviously, diam_{d} B_{d}(0, 1) ≤ 2; let ξ = (t, 0, …, 0) be such that d(ξ, 0) = 1, and notice that, by translation invariance, d(0, −ξ) = 1, so that ±ξ ∈ B_{d}(0, 1). Again by translation invariance d(−ξ, ξ) = d(0, ξ · ξ) = d(0, δ_{2}(ξ)) = 2d(0, ξ) = 2, where the equality ξ · ξ = δ_{2}(ξ) relies on the particular form of ξ.

To prove (ii), by homogeneity it is enough to show that the assertion holds for at least one r_{0} > 0. On the other hand, if μ(∂B_{d}(0, r)) > 0 for r > 0, then r is a discontinuity point for the increasing map r → μ(B_{d}(0, r)), and hence μ(∂B_{d}(0, r)) = 0 except for r in a countable set.

2.2. Calculus in Carnot groups. The following definitions and results about intrinsic differentiability in Carnot groups are basically due to P. Pansu ([56]), or are inspired by his ideas.

A map L : 𝔾 → ℝ is 𝔾-linear if it is a homomorphism from 𝔾 ≡ (ℝ^{n}, ·) to (ℝ, +) and if it is positively homogeneous of degree 1 with respect to the dilations of 𝔾, that is L(δ_{λ}x) = λLx for λ > 0 and x ∈ 𝔾. The ℝ-linear set of 𝔾-linear functionals 𝔾 → ℝ is indicated as L_{𝔾} and it is endowed with the norm

∥L∥_{L_{𝔾}} := sup{|L(p)| : d_{c}(p, 0) ≤ 1}.

Given a basis X_{1}, …, X_{n}, all 𝔾-linear maps are represented as the follows.

Proposition 2.5. A map L : 𝔾 → ℝ is 𝔾-linear if and only if there is a = (a_{1}, …, a_{m_{1}}) ∈ ℝ^{m_{1}} such that, if x = (x_{1}, …, x_{n}) ∈ 𝔾, then L(x) = ∑_{i=1}^{m_{1}} a_{i}x_{i}.

Proof. By (2.1), it is clear that any function L of the form L(v) = ∑_{i=1}^{m_{1}} a_{i}v_{i} is 𝔾-linear. Inversely, let L be 𝔾-linear and recall that

x = (x_{1}, …, x_{n}) = [x^{1}, …, x^{k}] ∈ ℝ^{n}

where

x^{j} = (x_{h_{j−1}+1}, …, x_{h_{j}}) ∈ ℝ^{m_{j}}, for 1 ≤ j ≤ k.

First we prove that x = [0, x^{2}, …, x^{k}] ⟹ Lx = 0. Indeed observe that if 1 < j ≤ k and only the elements of the j-th layer of x are non zero, i.e. x = [0, …, 0, x^{j}, 0, …, 0], then

2Lx = L(x · x) = L(δ_{2^{1/α_{j}}} x) = 2^{1/α_{j}}Lx

because x · x = [0, …, 0, 2x^{j}, 0, …, 0] = δ_{2^{1/α_{j}}} x, hence

x = [0, …, 0, x^{j}, 0, …, 0] ⟹ Lx = 0. (12)

Now remember that, from (6),

x = [0, x^{2}, …, x^{k}] = [0, x^{2}, 0, …, 0] · [0, 0, x^{3}, y^{4}, …, y^{k}]

for appropriate y^{4}, …, y^{k}. Hence, from (12) we have

L[0, x^{2}, …, x^{k}] = L[0, x^{2}, 0, …, 0] + L[0, 0, x^{3}, y^{4}, …, y^{k}] = L[0, 0, x^{3}, y^{4}, …, y^{k}].

Iterate this procedure to get, after k − 3 steps,

L[0, x^{2}, …, x^{k}] = L[0, …, 0, ỹ^{k}] = 0, (13)

from (12). Finally, x = [x^{1}, …, x^{k}] = [x^{1}, 0, …, 0] · [0, x^{2}, y^{3}, …, y^{k}], again for appropriate y^{3}, …, y^{k}; hence, from (13),

Lx = L[x^{1}, 0, …, 0]. (14)

Since 𝔾 ∩ {x^{2} = 0, …, x^{k} = 0} ≡ ℝ^{m_{1}} and L|_{𝔾∩{x^{2}=0,...,x^{k}=0}} : ℝ^{m_{1}} → ℝ is ℝ-linear, the thesis follows from the usual representation of linear functions on Euclidean spaces.

Definition 2.6. Let Ω be an open set in 𝔾, then f : Ω → ℝ is Pansu-differentiable (differentiable in the sense of Pansu: see [56] and [47]) at x_{0} if there is a 𝔾-linear map L such that

lim_{x→x_{0}} \frac{f(x) − f(x_{0}) − L(x_{0}^{−1} · x)}{d_{c}(x, x_{0})} = 0.

Remark 2.7. The above definition is equivalent to the following one: there exists a homomorphism L from 𝔾 to (ℝ, +) such that

lim_{λ→0+} \frac{f(τ_{x_{0}}(δ_{λ}v)) − f(x_{0})}{λ} = L(v)

uniformly with respect to v belonging to compact sets in 𝔾. In particular, L is unique and we shall write L = d_{𝔾}f(x_{0}). Notice that this definition of differential depends only on 𝔾 and not on the particular choice of the canonical generating vector fields. Indeed any two cc-distances induced by different choices of (equivalent) scalar products in H𝔾 are equivalent as distances.

Definition 2.8. If Ω is an open set in 𝔾, we denote by C_{𝔾}^{1}(Ω) the set of continuous real functions in Ω such that d_{𝔾}f : Ω → L_{𝔾} is continuous in Ω. Moreover, we shall denote by C_{𝔾}^{1}(Ω, H𝔾) the set of all sections ϕ of H𝔾 whose canonical coordinates ϕ_{j} ∈ C_{𝔾}^{1}(Ω) for j = 1, …, m_{1}.

Remark 2.9. We recall that C^{1}(Ω) ⊂ C_{𝔾}^{1}(Ω) and that the inclusion may be strict, for an example see Remark 6 in [32].

We say that f is differentiable along X_{j}, j = 1, …, m_{1}, at x_{0} if the map λ ↦ f(τ_{x_{0}}(δ_{λ}e_{j})) is differentiable at λ = 0, where e_{j} is the j-th vector of the canonical basis of ℝ^{n}.

Once a generating family of vector fields X_{1}, …, X_{m_{1}} is fixed, we define, for any function f : 𝔾 → ℝ for which the partial derivatives X_{j}f exist, the horizontal gradient of f, denoted by ∇_{𝔾}f, as the horizontal section

∇_{𝔾}f := ∑_{i=1}^{m_{1}}(X_{i}f)X_{i}.

whose coordinates are (X_{1}f, ..., X_{m_{1}}f). Moreover, if ϕ = (ϕ_{1}, …, ϕ_{m_{1}}) is an horizontal section such that X_{j}ϕ_{j} ∈ L_{loc}^{1}(𝔾) for j = 1, …, m_{1}, we define div_{𝔾} ϕ as the real valued function

div_{𝔾} (ϕ) := − ∑_{j=1}^{m_{1}} X_{j}^{∗}ϕ_{j} = ∑_{j=1}^{m_{1}} X_{j}ϕ_{j}.

Remark 2.10. The notation we have used for the gradient in a group is partially imprecise, indeed ∇_{𝔾}f really depends on the choice of the basis X_{1}, ..., X_{m_{1}}. If we choose a different base, say Y_{1}, ..., Y_{m_{1}}, then in general ∑_{i}(X_{i}f)X_{i} ≠ ∑_{i}(Y_{i}f)Y_{i}. Only if the two bases are one orthonormal with respect to the scalar product induced by the other, we have that

∑_{i}(X_{i}f)X_{i} = ∑_{i}(Y_{i}f)Y_{i}.

On the contrary, the notation div_{𝔾} used for the divergence is correct. Indeed div_{𝔾} is an intrinsic notion and it can be computed using the previous formula for any fixed generating family.

Finally, if x = (x_{1}, …, x_{n}) ∈ ℝ^{n} ≡ 𝔾 and x_{0} ∈ 𝔾 are given, we set

π_{x_{0}}(x) = ∑_{j=1}^{m_{1}} x_{j}X_{j}(x_{0}).

The map x_{0} → π_{x_{0}}(x) is a smooth section of H𝔾.

Proposition 2.11. If f is Pansu-differentiable at x_{0}, then it is differentiable along X_{j} at x_{0} for j = 1, …, m_{1}, and

d_{𝔾}f(x_{0})(v) = ⟨∇_{𝔾}f, π_{x_{0}}(v)⟩_{x_{0}}. (15)

For a proof see [55] Remark 3.3. The following proposition can be proved via an approximation argument as in Proposition 5.8 of [32].

Proposition 2.12. A continuous function belongs to C_{𝔾}^{1}(Ω) if and only if its distributional derivatives X_{j}f are continuous in Ω for j = 1, …, m_{1}.

Remark 2.13. As we observed both ∇_{𝔾} and the cc-distance d_{c} depend on the choice of the canonical generating family X_{j}. But the eikonal equation connecting the two notions

|∇_{𝔾}d_{c}(0, x)| = 1 (16)

holds for L^{n}-a.e. x ∈ 𝔾 and for all generating family (see Theorem 3.1 of [55]).

An extension theorem of Whitney type holds. Its rather lengthy proof is postponed to the Appendix.

Theorem 2.14. (Whitney Extention Theorem) Let F ⊂ 𝔾 be a closed set, and let f : F → ℝ, k : F → H𝔾 be, respectively, a continuous real function and a continuous horizontal section. We set

R(x, y) := \frac{f(x) − f(y) − ⟨k(y), π_{y}(y^{−1} · x)⟩_{y}}{d_{c}(y, x)} ,

and, if K ⊂ F is a compact set,

ρ_{K}(δ) := sup{|R(x, y)| : x, y ∈ K, 0 < d_{c}(x, y) < δ}.

Assume

ρ_{K}(δ) → 0 as δ → 0 for every compact set K ⊂ F,

then there exist f̃: 𝔾 → ℝ, f̃ ∈ C_{𝔾}^{1}(𝔾) such that

f̃_{|F} = f, ∇_{𝔾}f̃_{|F} = k.

2.3. BV -functions and finite perimeter sets. As for the Euclidean theory of BV functions and finite perimeter sets, the reader can see, for instance, [3], [25], [40], and [68]. The definitions of BV functions and of finite perimeter sets in Carnot groups, were given explicitly for the first time in [12]. For more details and proofs, see [37] and also [29]. Fine properties of BV functions in Carnot groups are deeply analysed in [6].

If Ω ⊆ ℝ^{n} is open, the space of compactly supported smooth sections of H𝔾 is denoted by C_{0}^{∞}(Ω, H𝔾). If k ∈ ℕ, C_{0}^{k}(Ω, H𝔾) is defined analogously.

The space BV_{𝔾}(Ω) is the set of functions f ∈ L^{1}(Ω) such that

(17) ||∇_{𝔾}f||(Ω) := sup {∫_{Ω} f(x)div_{𝔾} ϕ(x) dx : ϕ ∈ C_{0}^{1}(Ω, H𝔾), |ϕ(x)|_{x} ≤ 1} < ∞.

The space BV_{𝔾,loc}(Ω) is the set of functions belonging to BV_{𝔾}(U) for each open set U ⊂⊂ Ω. Notice the use of the intrinsic fiber norm inside the previous definition.

Theorem 2.15. (Structure of BV_{𝔾} functions) If f ∈ BV_{𝔾,loc}(Ω) then ||∇_{𝔾}f|| is a Radon measure on Ω. Moreover, there exists a ||∇_{𝔾}f||-measurable horizontal section σ_{f} : Ω → H𝔾 such that |σ_{f}(x)|_{x} = 1 for ||∇_{𝔾}f||-a.e. x ∈ Ω, and

∫_{Ω} f(x)div_{𝔾} ϕ(x) dx = ∫_{Ω}⟨ϕ, σ_{f}⟩ d||∇_{𝔾}f||,

for all ϕ ∈ C_{0}^{1}(Ω, H𝔾). Finally the notion of gradient ∇_{𝔾} can be extended from regular functions to functions f ∈ BV_{𝔾} defining ∇_{𝔾}f as the vector valued measure

∇_{𝔾}f := −σ_{f} ||∇_{𝔾}f|| = (−(σ_{f})_{1} ||∇_{𝔾}f||, …, −(σ_{f})_{m_{1}} ||∇_{𝔾}f||) ,

where (σ_{f})_{j} are the components of σ_{f} with respect to the moving base X_{j}.

It is well known that the usefulness of these definitions for the Calculus of Variations, relies mainly in the validity of the two following theorems. In the context of subriemannian geometries they are proved respectively in [37] and [29].

Theorem 2.16. (Compactness) BV_{𝔾,loc}(𝔾) is compactly embedded in L_{loc}^{p}(𝔾) for 1 ≤ p < \frac{Q}{Q−1} where Q, defined in (10), is the homogeneous dimension of 𝔾.

Theorem 2.17. (Lower semicontinuity) Let f, f_{k} ∈ L^{1}(Ω), k ∈ ℕ, be such that f_{k} → f in L^{1}(Ω); then

lim inf_{k→∞} ||∇_{𝔾}f_{k}||(Ω) ≥ ||∇_{𝔾}f||(Ω).

Definition 2.18. A measurable set E ⊂ ℝ^{n} is of locally finite 𝔾-perimeter in Ω (or is a 𝔾-Caccioppoli set) if the characteristic function 1_{E} ∈ BV_{𝔾,loc}(Ω). In this case we call perimeter of E the measure

|∂E|_{𝔾} := ||∇_{𝔾}1_{E}|| (18)

and we call (generalized inward) 𝔾-normal to ∂E in Ω the vector

ν_{E}(x) := −σ_{1_{E}}(x). (19)

Remark 2.19. This remark is analogous to remark 2.10. The symbol |∂E|_{𝔾} is somehow incorrect, indeed the value of the 𝔾-perimeter depends on the choice of the generating vector fields X_{1}, …, X_{m_{1}}, precisely through the bound |ϕ| ≤ 1 in (17). The values of the perimeters induced by two different families of generating vector fields coincide only if the two families are mutually orthonormal; nevertheless the perimeters induced by different families are equivalent as measures and, as a consequence, the notion of being a 𝔾-Caccioppoli set is an intrinsic one depending only on the group 𝔾.

Remark 2.20. The 𝔾-perimeter is invariant under group translations, that is

|∂E|_{𝔾}(A) = |∂(τ_{p}E)|_{𝔾}(τ_{p}A), ∀p ∈ 𝔾, and for any Borel set A ⊂ 𝔾;

indeed div_{𝔾} is invariant under group translations and the Jacobian determinant of τ_{p} : 𝔾 → 𝔾 equals 1. Moreover the 𝔾-perimeter is homogeneous of degree Q−1 with respect to the dilations of the group, that is

(20) |∂(δ_{λ}E)|_{𝔾}(A) = λ^{1−Q}|∂E|_{𝔾}(δ_{λ}A), for any Borel set A ⊂ 𝔾;

also this fact is elementary and can be proved by changing variables in formula (17).

Lemma 2.21. (Localization) Let E be a 𝔾-Caccioppoli set, x ∈ 𝔾 and

m(ρ) := L^{n}(E ∩ U_{c}(x, ρ)).

Then, for L^{1}-a.e. ρ > 0, E ∩ U_{c}(x, ρ) is a 𝔾-Caccioppoli set,

|∂(E ∩ U_{c}(x, ρ))|_{𝔾}(ℝ^{n}) ≤ |∂E|_{𝔾}(U_{c}(x, ρ)) + m′(ρ), (21)

and

|∂(E ∩ U_{c}(x, ρ))|_{𝔾}(∂U_{c}(x, ρ)) ≤ m′(ρ). (22)

Proof. Choose ρ > 0 such that

|∂E|_{𝔾}(∂U_{c}(x, ρ)) = 0. (23)

Observe that this can be done for L^{1}-a.e. ρ > 0. Estimate (21) follows if we can show that for σ ∈ (0, +∞) there are approximating functions u_{σ} ∈ BV_{𝔾}(ℝ^{n}) such that u_{σ} → 1_{E∩U_{c}(0,R)} in L^{1}(ℝ^{n}), for σ → 0, u_{σ} is supported in \overline{U_{c}(x, ρ + σ)}, u_{σ} = 1_{E} on U_{c}(x, ρ) and

∥∇_{𝔾}u_{σ}∥(ℝ^{n}) ≤ |∂E|_{𝔾}(U_{c}(x, ρ + σ)) + \frac{1}{σ} ∫_{U_{c}(x,ρ+σ)\U_{c}(x,ρ)} 1_{E}dL^{n}. (24)

Indeed, as σ → 0, by the lower semicontinuity of the variation ∥∇_{𝔾}u_{σ}∥, (24) gives

|∂(E ∩ U_{c}(x, ρ))|_{𝔾}(ℝ^{n}) ≤ |∂E|_{𝔾}(\overline{U_{c}(x, ρ)}) + m′(ρ),

and the thesis follows from (23). Notice that we use here that ∂B(x, R) = {y : d_{c}(x, y) = R} as is proved in Proposition 2.4. The construction of u_{σ} can be obtained following with minor modifications the one in Proposition 3.56 in [3], given that |∇_{𝔾}d_{c}(x, y)|_{y} = 1 for L^{n}-a.e. y ∈ 𝔾 (see (16)).

The second statement (22) follows from (21), by the additivity of measures and because of the following general property of perimeters

(25)  |∂(E ∩ U)|_{𝔾}(U) = |∂E|_{𝔾}(U),  for all U open, U ⊂ ℝ^{n}.

The following two theorems were proved respectively in [12] and [37].

Proposition 2.22. If E is a 𝔾-Caccioppoli set with Euclidean C^{1} boundary, then there is an explicit representation of the 𝔾-perimeter in terms of the Euclidean (n−1)-dimensional Hausdorff measure H^{n−1}

|∂E|_{𝔾}(Ω) = ∫_{∂E∩Ω} ( ∑_{j=1}^{m_{1}}⟨X_{j}, n⟩_{ℝ^{n}}^{2})^{1/2}dH^{n−1},

where n = n(x) is the Euclidean unit outward normal to ∂E.

Metric balls in 𝔾 satisfy a relative isoperimetric inequality, that is

Proposition 2.23. (Isoperimetric inequality) There is a positive constant c_{I} > 0 such that for any 𝔾-Caccioppoli set E, for all x ∈ 𝔾 and r > 0,

min\{L^{n}(E ∩ U_{c}(x, r)), L^{n}(E^{c} ∩ U_{c}(x, r))\}^{\frac{Q−1}{Q}} ≤ c_{I}|∂E|_{𝔾}(U_{c}(x, r)) (26)

and

min\{L^{n}(E), L^{n}(E^{c})\}^{\frac{Q−1}{Q}} ≤ c_{I}|∂E|_{𝔾}(ℝ^{n}). (27)

If E ⊂ 𝔾, as it is usual in the literature, we define the measure theoretic boundary ∂_{∗,𝔾}E – called also essential boundary – and the reduced boundary ∂_{𝔾}^{∗}E.

Definition 2.24. (Measure theoretic boundary) Let E ⊂ 𝔾 be a measurable set, we say that x ∈ ∂_{∗,𝔾}E if

lim sup_{r→0^{+}} \frac{L^{n}(E ∩ U_{c}(x, r))}{L^{n}(U_{c}(x, r))} > 0  and  lim sup_{r→0^{+}} \frac{L^{n}(E^{c} ∩ U_{c}(x, r))}{L^{n}(U_{c}(x, r))} > 0.

Definition 2.25. (Reduced boundary) Let E be a 𝔾-Caccioppoli set; we say that x ∈ ∂_{𝔾}^{∗}E if

|∂E|_{𝔾}(U_{c}(x, r)) > 0 for any r > 0; (i)

(ii) there exists lim_{r→0} ∫_{U_{c}(x,r)} ν_{E} d|∂E|_{𝔾};

∥lim_{r→0} ∫_{U_{c}(x,r)} ν_{E} d|∂E|_{𝔾}∥_{ℝ^{m_{1}}} = 1. (iii)

The limits in Definition 2.25 should be understood as a convergence of the averages of the coordinates of ν_{E} with respect to the chosen moving base of the fibers. The same convention will be used throughout all paper whenever integrations of the normals are involved.

Some more comments about these two definitions are useful. Before them we state the following differentiation lemma that will play a key role in the present paper. Observe that this differentiation Lemma, in the Euclidean setting, is an immediate consequence of Lebesgue-Besicovitch differentiation Lemma, while in Carnot groups (where Lebesgue-Besicovitch fails to hold, see [47] and [60]) it relies on a deep asymptotic estimate proved by Ambrosio in [1].

Lemma 2.26. (Differentiation Lemma) Assume E is a 𝔾-Caccioppoli set, then

lim_{r→0} ∫_{U_{c}(x,r)} ν_{E} d|∂E|_{𝔾} = ν_{E}(x), for |∂E|_{𝔾}-a.e. x,

that is |∂E|_{𝔾}-a.e. x ∈ 𝔾 belongs to the reduced boundary ∂_{𝔾}^{∗}E.

Proof. The proof is the same as in Lemma 5 of [32]. From Corollary 4.5 of [1]

(28)  lim sup_{r→0} \frac{|∂E|_{𝔾}(B_{c}(x, 2r))}{|∂E|_{𝔾}(B_{c}(x, r))} < ∞  for |∂E|_{𝔾}-a.e. x ∈ 𝔾 .

Now we apply 2.8.17 in [26] with X := 𝔾, V (X) := {B_{c}(x, r) : x ∈ X, r > 0}, δ(B_{c}(x, r)) := r/4, τ := 2, Φ := |∂E|_{𝔾} and

B̃_{c}(x, r) := ⋃ {B_{c}(y, ρ) : ρ < r, B_{c}(y, ρ) ∩ B_{c}(x, r) ≠ ∅} ⊆ B_{c}(x, 2r),

getting that V (𝔾) is a |∂E|_{𝔾}-Vitali relation (see [26], 2.8.16). Then, from 2.9.8 in [26] with f = ν_{E}, we get

(29)  lim_{r→0} ∫_{B_{c}(x,r)} ν_{E} d|∂E|_{𝔾} = ν_{E}(x),  for |∂E|_{𝔾}-a.e. x ∈ 𝔾.

Finally, by usual arguments of measure theory, one can replace the closed balls B_{c}(x, r) in (29) with the open ones U_{c}(x, r).

Remark 2.27. We shall prove below that, as a consequence of the existence of a generalized tangent group in the points of the reduced boundary (Theorem 3.1), ∂_{∗,𝔾}E ⊃ ∂_{𝔾}^{∗}E, as is true in the Euclidean setting.

Remark 2.28. Two equivalent but different doubling distances on 𝔾 yield the same measure theoretic boundary of E while the respectively induced reduced boundaries can be different. This is not a pointless observation since there are several different natural distances (equivalent to d_{c}) on any Carnot group; later on we shall use one of these: the d_{∞} distance defined in (8), to get an explicit representation of the perimeter measure in terms of a suitable Hausdorff measure.

Still we notice that Lemma 2.26 holds also if we replace Carnot-Carathéodory balls with balls coming from an equivalent metric. As a consequence the reduced boundaries associated with equivalent distances (and equivalent to d_{c}) coincide up to a set of vanishing 𝔾-perimeter.

Remark 2.29. The definition of ∂_{𝔾}^{∗}E depends only on E and not on the particular representative ν_{E} since only integrals of it are involved. Moreover, because of differentiation Lemma 2.26 the perimeter measure is concentrated on the reduced boundary and possibly after redefining ν_{E} in a |∂E|_{𝔾}-negligible set, we can assume that ν_{E} is the limit of the averages at all point x ∈ ∂_{𝔾}^{∗}E.

Remark 2.30. The 𝔾-reduced boundary of a set is invariant under group translations, i.e.

x_{0} ∈ ∂_{𝔾}^{∗}E if and only if τ_{x}x_{0} ∈ ∂_{𝔾}^{∗}(τ_{x}E)

and also

ν_{E}(x_{0}) = ν_{τ_{x}E}(τ_{x}x_{0}).

The following Lemma gives preliminary informations about the regularity of the boundary of a finite perimeter set near the points of its reduced boundary. It will be an essential tool in the proof of the rectifiability of the reduced boundary.

Lemma 2.31. (Density Estimates) Let E be a Caccioppoli set in 𝔾 and x ∈ ∂_{𝔾}^{∗}E. Then there exist three positive constants c_{i} = c_{i}(𝔾) > 0, i = 1, 2 and ρ_{0} = ρ(x) > 0 such that

|∂E|_{𝔾}(U_{c}(x, ρ)) ≤ c_{1}ρ^{Q−1}, 0 < ρ < ρ_{0} (30)

L^{n}(E ∩ U_{c}(x, ρ)) ≥ c_{2}ρ^{Q}, 0 < ρ < ρ_{0}. (31)

Proof. Because x ∈ ∂_{𝔾}^{∗}E, there is ρ_{0} > 0 such that

|∂E|_{𝔾}(U_{c}(x, ρ)) ≤ 2||∇_{𝔾}1_{E}(U_{c}(x, ρ))||_{R^{m_{1}}} , for all ρ ≤ 2ρ_{0}. (32)

For a.e. ρ ∈ (0, 2ρ_{0}) both |∂(E ∩ U_{c}(x, ρ))|_{𝔾} < ∞ and (22) holds. Hence, for these ρ we have

∇_{𝔾}1_{E∩U_{c}(x,ρ)}(U_{c}(x, ρ)) = −∇_{𝔾}1_{E∩U_{c}(x,ρ)}(∂U_{c}(x, ρ)), (33)

because ∇_{𝔾}1_{E∩U_{c}(x,ρ)}(ℝ^{n}) = 0 and ∇_{𝔾}1_{E∩U_{c}(x,ρ)}(ℝ^{n} \ \overline{U_{c}(x, ρ)}) = 0. Then, from (25), (33) and (22),

||∇_{𝔾}1_{E}(U_{c}(x, ρ))||_{ℝ^{m_{1}}} = ||∇_{𝔾}1_{E∩U_{c}(x,ρ)}(U_{c}(x, ρ))||_{ℝ^{m_{1}}} = ||∇_{𝔾}1_{E∩U_{c}(x,ρ)}(∂U_{c}(x, ρ))||_{ℝ^{m_{1}}}
= |∂(E ∩ U_{c}(x, ρ))|_{𝔾}(∂U_{c}(x, ρ))
≤ m′(ρ).

Hence from (32) it follows

|∂E|_{𝔾}(U_{c}(x, ρ)) ≤ 2m′(ρ), for a.e. ρ ∈ (0, 2ρ_{0}). (34)

Now integrate (34) from ρ to 2ρ, use the fact that |∂E|_{𝔾}(U_{c}(x, ρ)) is non decreasing and that m′(ρ) > 0, to get

|∂E|_{𝔾}(U_{c}(x, ρ)) ≤ 2\frac{m(2ρ)}{ρ} ≤ 2^{Q+1}L^{n}(U_{c}(0, 1))ρ^{Q−1}

whence (30) with c_{1} = 2^{Q+1}L^{n}(U_{c}(0, 1)). this completes the proof of (30).

Finally from isoperimetric inequality (2.23) and the previous Localization Lemma we get, for almost every ρ ∈ (0, 2ρ_{0}),

m(ρ)^{\frac{Q−1}{Q}} ≤ c_{I}|∂(E ∩ U_{c}(x, ρ))|_{𝔾}(ℝ^{n})
≤ c_{I} (|∂E|_{𝔾}((∩U_{c}(x, ρ))) + m′(ρ))
≤ 3c_{I}m′(ρ),

hence

(m(ρ)^{\frac{1}{Q}})′ ≥ \frac{1}{3c_{I}Q}, for a.e. ρ ∈ (0, 2ρ_{0})

and then

m(ρ) ≥ c_{2}ρ^{Q} ∀ρ ∈ (0, 2ρ_{0}).

2.4. Regular hypersurfaces, rectifiable sets, and Implicit Function Theorem. Following [33], we define 𝔾-regular hypersurfaces in a Carnot group 𝔾 as non critical level sets of functions in C_{𝔾}^{1}(ℝ^{n}).

Definition 2.32. S ⊂ 𝔾 is a 𝔾-regular hypersurface if for every x ∈ S there exist a neighborhood U of x and a function f ∈ C_{𝔾}^{1}(U) such that

S ∩ U = {y ∈ U : f(y) = 0}; (i) ∇_{𝔾}f(y) ≠ 0 for y ∈ U. (ii)

Definition 2.33. Γ ⊂ 𝔾 is said to be ( (Q − 1)-dimensional) 𝔾-rectifiable if there exists a sequence of 𝔾-regular hypersurfaces (S_{j})_{j∈ℕ} such that

H_{c}^{Q−1} (Γ \ ⋃_{j∈ℕ}S_{j}) = 0. (35)

If S = {x : f(x) = 0} ⊂ 𝔾 is a 𝔾-regular hypersurface, the tangent group T_{𝔾}^{g}S(x) to S at x is

T_{𝔾}^{g}S(x) := {v ∈ 𝔾 : ⟨∇_{𝔾}f(x), π_{x}v⟩_{x} = 0}.

By (2.1), T_{𝔾}^{g}S(x) is a proper subgroup of 𝔾. We can define the tangent plane to S at x as

T_{𝔾}S(x) := x · T_{𝔾}^{g}S(x).

We stress that this is a good definition. Indeed the tangent plane does not depend on the particular function f defining the surface S because of point (iii) of Implicit Function Theorem below that yields

T_{𝔾}^{g}S(x) = {v ∈ 𝔾 : ⟨ν_{E}(x), π_{x}v⟩_{x} = 0}

where ν_{E} is the generalized inward unit normal defined in (19). Notice also that, once more from (iii) of Theorem 2.35, it follows that ν_{E} is a continuous function.

Note also that the class of 𝔾-regular hypersurfaces is different from the class of Euclidean C^{1} embedded surfaces in ℝ^{n}. From one side 𝔾-regular surfaces can have ’ridges’ because continuity of the derivatives of the defining functions f is required only in the horizontal directions; on the other side an Euclidean C^{1} surface can have so called characteristic points i.e. points p ∈ S where the Euclidean tangent plane T_{p}S contains the horizontal fiber H𝔾_{p}.

Definition 2.34. If S is an Euclidean C^{1} hypersurface in 𝔾, we define the characteristic set of S as

C(S) := {x ∈ S : H𝔾_{x} ⊆ T_{x}S}. (36)

The points of C(S) are, under many aspects, irregular points of S. Note indeed that the tangent group does not exist in these points. It is also well known that these points are ’few’ on smooth hypersurfaces, but only recently Balogh [7] obtained precise estimates of the H_{c}^{Q−1} measure of the characteristic sets of C^{1} surfaces in the Heisenberg groups ℍ^{n}. For Euclidean C^{1,1} surfaces in Carnot groups of step 2 see also [49]. In the last section we extend Balogh’s result to C^{1} hypersurfaces in step 2 Carnot groups.

The following Implicit Function Theorem states that any 𝔾-regular hypersurface S = {y ∈ 𝔾 : f(y) = 0}, boundary of the set E = {y ∈ 𝔾 : f(y) < 0}, can be locally parameterized through a function Φ : ℝ^{n−1} → ℝ^{n} so that the 𝔾-perimeter of E can be written explicitly in terms of ∇_{𝔾}f and of Φ.

Theorem 2.35. (Implicit Function Theorem) Let Ω be an open set in ℝ^{n}, 0 ∈ Ω, and let f ∈ C_{𝔾}^{1}(Ω) be such that f(0) = 0 and X_{1}f(0) > 0. Define

E = {x ∈ Ω : f(x) < 0}, S = {x ∈ Ω : f(x) = 0},

and, for δ > 0, h > 0

I_{δ} = {ξ = (ξ_{2}, …, ξ_{n}) ∈ ℝ^{n−1}, |ξ_{j}| ≤ δ}, J_{h} = [−h, h].

If ξ = (ξ_{2}, …, ξ_{n}) ∈ ℝ^{n−1} and t ∈ J_{h}, denote by γ(t, ξ) the integral curve of the vector field X_{1} at the time t issued from (0, ξ) = (0, ξ_{2}, …, ξ_{n}) ∈ ℝ^{n}, i.e.

γ(t, ξ) = exp(tX_{1})(0, ξ).

Then there exist δ, h > 0 such that the map (t, ξ) → γ(t, ξ) is a homeomorphism of a neighborhood of J_{h} × I_{δ} onto an open subset of ℝ^{n}, and, if we denote by U ⊂⊂ Ω the image of Int(J_{h} × I_{δ}) through this map, we have

(i) E has finite 𝔾-perimeter in U; ∂E ∩ Ω = S ∩ U; (ii)

ν_{E}(x) = − \frac{∇_{𝔾}f(x)}{|∇_{𝔾}f(x)|_{x}} for all x ∈ S ∩ Ω, (iii)

where ν_{E} is the generalized inner unit normal defined by (19). Moreover, there exists a unique function

ϕ = ϕ(ξ) : I_{δ} → J_{h}

such that the following parameterization holds: if ξ ∈ I_{δ} and we put Φ(ξ) = γ(ϕ(ξ), ξ), then

S ∩ Ω̃ = {x ∈ Ũ : x = Φ(ξ), ξ ∈ I_{δ}}; (iv)

(v) ϕ is continuous;

the 𝔾-perimeter has an integral representation

|∂E|_{𝔾}(Ũ) = ∫_{I_{δ}} \frac{\sqrt{∑_{j=1}^{m} |X_{j}f(Φ(ξ))|^{2}}}{X_{1}f(Φ(ξ))} dξ. (vi)

3. Structure of Finite Perimeter Sets in Step 2 Groups.

From now on the group 𝔾 will be a step 2 Carnot group. Indeed, the keystep for the main result of this paper, i.e. the so-called Blow-up Theorem stated below, fails to be true for general groups of step greater than 2, as we can see from the Example 1.

Specializing our notations, in step 2 Carnot groups, we have

𝔤 = V_{1} ⊕ V_{2}, [V_{1}, V_{1}] = V_{2}, [V_{1}, V_{2}] = {0},

and

Q = m_{1} + 2(n − m_{1}).

The main results of the present section are:

i) At each point of the reduced boundary of a 𝔾-Caccioppoli set there is a (generalized) tangent group; ii) Both the reduced boundary and the measure-theoretic boundary are (Q − 1)-dimensional 𝔾-rectifiable sets; iii) |∂E|_{𝔾} = c S_{∞}^{Q−1} ∂^{∗}E, i.e. the perimeter measure equals a constant times the spherical (Q − 1)-dimensional Hausdorff measure restricted to the reduced boundary. iv) An intrinsic divergence theorem holds for 𝔾-Caccioppoli sets.

The precise meaning of statement i) is the content of the following Blow-up Theorem. It is precisely point i) that can be false in a general Carnot group. Indeed we provide an example of a 𝔾-regular hypersurface S = ∂E in a step 3 group (the so-called Engel group, see e.g. [41], [53]) such that 0 ∈ ∂_{𝔾}^{∗}E but E has not generalized tangent group at that point.

Statement iii) fits in the general problem of comparing different geometric measures in Carnot groups. A good reference for this problem, in Euclidean spaces, is Matilla’s book [50]. In the setting of the Heisenberg group, in [16] it is proved that the perimeter of an Euclidean C^{1,1}- hypersurface is equivalent to its (Q−1)-dimensional intrinsic Hausdorff measure, whereas in [32] it is proved that on the boundary of sets of finite intrinsic perimeter the (Q − 1)-dimensional intrinsic spherical Hausdorff measure coincide – after a suitable normalization – with the perimeter measure. In the setting of general Carnot groups the problem is essentially open. The equivalence of the intrinsic perimeter and of the (Q − 1)-dimensional intrinsic Hausdorff measure for C_{𝔾}^{1}-hypersurfaces in a general Carnot groups is proved in [33]. In addition, the perimeter measure in general subriemannian spaces equals the intrinsic Minkowski content, as it is proved in [55]. In Ahlfors-regular metric spaces, a general representation theorem of the perimeter measure of sets of finite perimeter in terms of the Hausdorff measure is proved in [1] (see also the refined result for subriemannian manifolds in [2]), showing that the intrinsic perimeter admits a density θ with respect to the Hausdorff measure that is locally summable and bounded away from zero. Statement iii) says precisely that, thanks to i) and ii), in step 2 Carnot groups the function θ is constant.

To state our result, let us fix few notations. For any set E ⊂ 𝔾, x_{0} ∈ 𝔾 and r > 0 we consider the translated and dilated sets E_{r,x_{0}} defined as

E_{r,x_{0}} = \{x : x_{0} · δ_{r}(x) ∈ E\} = δ_{\frac{1}{r}} τ_{x_{0}^{−1}}E.

If x_{0} is fixed and there is no ambiguity, we shall write simply E_{r}, and in addition we set E_{x_{0}} = E_{1,x_{0}}. Moreover if v ∈ H𝔾_{x_{0}} we define the halfspaces S_{𝔾}^{+}(v) and S_{𝔾}^{−}(v) as

\begin{aligned}S_{𝔾}^{+}(v) & := \{x : ⟨π_{x_{0}}x, v⟩_{x_{0}} ≥ 0\} \\ S_{𝔾}^{−}(v) & := \{x : ⟨π_{x_{0}}x, v⟩_{x_{0}} ≤ 0\}. \end{aligned} (37)

The common topological boundary T_{𝔾}^{g}(v) of S_{𝔾}^{+}(v) and of S_{𝔾}^{−}(v) is the subgroup of 𝔾

T_{𝔾}^{g}(v) := {x : ⟨π_{x_{0}}x, v⟩_{x_{0}} = 0}.

Theorem 3.1. (Blow-up Theorem) If E is a 𝔾-Caccioppoli set, x_{0} ∈ ∂_{𝔾}^{∗}E and ν_{E}(x_{0}) ∈ H𝔾_{x_{0}} is the inward normal as defined in (19) then

(38)  lim_{r→0} 1_{E_{r,x_{0}}} = 1_{S_{𝔾}^{+}(ν_{E}(x_{0}))}  in L_{loc}^{1}(𝔾)

and for all R > 0

lim_{r→0} |∂E_{r,x_{0}}|_{𝔾}(U_{c}(0, R)) = |∂S_{𝔾}^{+}(ν_{E}(x_{0}))|_{𝔾}(U_{c}(0, R)). (39)

Notice that, by Proposition 2.22,

|∂S_{𝔾}^{+}(ν_{E}(x_{0}))|_{𝔾}(U_{c}(0, R)) = H^{n−1}(T_{𝔾}^{g}(ν_{E}(0)) ∩ U_{c}(0, R)).

We postpone the proof of Theorem 3.1 and we provide an example showing that the above statement does not hold any longer in groups of step k > 2.

The core of the following example consists in showing that in Carnot groups of step greater than 2 can exists cones (i.e. dilation-invariant sets) that are not flat (they are not of the form S_{𝔾}^{±}(v) for some horizontal vector v) but nevertheless with vertex belonging to the reduced boundary.

Example 1. Let us recall the definition of Engel algebra and group. Let 𝔼 = (ℝ^{4}, ·) be the Carnot group whose Lie algebra is 𝔤 = V_{1} ⊕ V_{2} ⊕ V_{3} with V_{1} = span {X_{1}, X_{2}}, V_{2} = span {X_{3}}, and V_{3} = span {X_{4}}, the only non zero commutation relations being

[X_{1}, X_{2}] = −X_{3} , [X_{1}, X_{3}] = −X_{4}.

In exponential coordinates the group law takes the form

x · y = H(∑_{i=1}^{4} x_{i}X_{i}, ∑_{i=1}^{4} y_{i}X_{i}),

where H is given by the Campbell-Hausdorff formula

H(X, Y) = X + Y + \frac{1}{2}[X, Y] + \frac{1}{12} ([X, [X, Y]] − [Y, [X, Y]]) . (40)

In exponential coordinates an explicit representation of the vector fields is

X_{1} = ∂_{1} + \frac{x_{2}}{2} ∂_{3} + (\frac{x_{3}}{2} − \frac{x_{1}x_{2}}{12} )∂_{4} , X_{2} = ∂_{2} − \frac{x_{1}}{2} ∂_{3} + \frac{x_{1}^{2}}{12}∂_{4}
X_{3} = ∂_{3} − \frac{x_{1}}{2} ∂_{4} , X_{4} = ∂_{4}.

Let E = {x ∈ ℝ^{4} : f(x) ≥ 0}, where

f(x) = \frac{1}{6}x_{2}(x_{1}^{2} + x_{2}^{2}) − \frac{1}{2}x_{1}x_{3} + x_{4}.

Since ∂E = {x ∈ ℝ^{4} : f(x) = 0} is a smooth Euclidean manifold, then E is a 𝔾-Caccioppoli set (see Proposition 2.22). Moreover,

∇_{𝔼}f(x) = (0, \frac{1}{2}(x_{1}^{2} + x_{2}^{2})),

and, by the Implicit Function Theorem (Theorem 2.35),

ν_{E}(x) = −\frac{∇_{𝔼}f(x)}{|∇_{𝔼}f(x)|} = (0, −1)

for all x ∈ ∂E \ N, where N = {x ∈ 𝔼 : x_{1} = x_{2} = 0}. Since |∂E|_{𝔼}(N) = 0, then the origin belongs to the reduced boundary of E. On the other hand, since f(δ_{λ}x) = λ^{3}f(x) for λ > 0, it follows that E_{λ,0} = δ_{λ}E = E, so that (38) fails to be true since E is not a vertical halfspace.

Proof of Blow up Theorem 3.1. Without loss of generality, it is possible to assume that x_{0} = 0 ∈ ∂_{𝔾}^{∗}E and we will drop the index x_{0} in our notation. To prove the theorem it is enough to show that, given any sequence {r_{k}}_{k} with r_{k} → 0, there exists a subsequence {s_{j}}_{j} ⊂ {r_{k}}_{k} such that

lim_{j→+∞} 1_{E_{s_{j}}} = 1_{S_{𝔾}^{+}(ν_{E}(0))}

in L_{loc}^{1}(𝔾) and that ∀R > 0

lim_{j→+∞} |∂E_{s_{j}}|_{𝔾}(U_{c}(0, R)) = H^{n−1}(T_{𝔾}^{g}(ν_{E}(0)) ∩ U_{c}(0, R));

notice the use of the Euclidean Hausdorff measure in the preceding formula.

The proof of these facts is divided in a number of steps. Claim 1. There is a 𝔾-Caccioppoli set F such that

(41)  lim_{j→+∞} 1_{E_{s_{j}}} = 1_{F} , in L_{loc}^{1}(𝔾).

Proof of Claim 1: Fix R > 0, then the homogeneity of the perimeter (see (20)) gives

|∂E_{r}|_{𝔾}(U_{c}(0, R)) = \frac{|∂E|_{𝔾}(U_{c}(0, rR))}{r^{Q−1}}, ∀r > 0. (42)

Because of lemma 2.31 the right hand side is bounded by a constant independent on r when 0 < r < ρ_{0}/R. It follows that for all R > 0 and 0 < r < ρ_{0}/R

|∂E_{r}|_{𝔾}(U_{c}(0, R)) ≤ c_{1}.

Moreover

∥1_{E_{r}}∥_{L^{1}(U_{c}(0,R))} = L^{n}(E_{r} ∩ U_{c}(0, R)) ≤ L^{n}(U_{c}(0, R)),

hence by the compactness theorem 2.16, there exist a sequence s_{j} → 0 and a function f ∈ BV_{𝔾,loc}(𝔾) such that

1_{E_{s_{j}}} → f, in L_{loc}^{1}(𝔾).

Passing to a subsequence, we may assume that the above sequence converges pointwise L^{n}-a.e. to f, and therefore f = 1_{F} for a suitable measurable set F ⊂ 𝔾. By the semicontinuity of the variation (see Theorem 2.17),

|∂F |_{𝔾}(U_{c}(0, R)) ≤ lim inf_{j→∞} |∂E_{s_{j}}|_{𝔾}(U_{c}(0, R)) ≤ c_{1} < ∞

for every R > 0. Hence F is a 𝔾-Caccioppoli set, i.e. 1_{F} ∈ BV_{𝔾,loc}(𝔾). In particular, by Theorem 2.15, for each ϕ ∈ C_{0}^{1}(𝔾, H𝔾)

− ∫_{F} div_{𝔾} ϕ dx = ∫_{𝔾} ⟨ν_{F} , ϕ⟩ d|∂F |_{𝔾}. (43)

This concludes the proof of Claim 1.

Claim 2. ν_{F} is a left invariant vector field, precisely

ν_{F} (x) = ν_{E}(0) for |∂F |_{𝔾}-a.e. x,

where the identity means, as usual, equality of the coordinates with respect to the moving base X_{j}. In particular this implies that F is independent of the sequence {s_{j}}_{j}.

Proof of Claim 2. Because 1_{E_{s_{j}}} → 1_{F} in L_{loc}^{1}(𝔾) as j → +∞, we have

lim_{j→+∞} ∫_{E_{s_{j}}} div_{𝔾} ϕ dx = ∫_{F} div_{𝔾} ϕ dx

for any ϕ ∈ C_{0}^{1}(𝔾, H𝔾) and, since both E_{s_{j}} and F are 𝔾-Caccioppoli sets,

lim_{j→+∞} ∫_{𝔾} ⟨ϕ, ν_{E_{s_{j}}} ⟩ d|∂E_{s_{j}}|_{𝔾} = ∫_{𝔾} ⟨ϕ, ν_{F} ⟩ d|∂F |_{𝔾}

that is

ν_{E_{s_{j}}} |∂E_{s_{j}}|_{𝔾} ⇀ ν_{F} |∂F |_{𝔾}, as j → +∞,

weakly as vector valued Radon measures. It follows that for any Borel set B ⊂ 𝔾 for which |∂F |_{𝔾}(∂B) = 0

lim_{j→+∞} ∫_{B} ν_{E_{s_{j}}} d|∂E_{s_{j}}|_{𝔾} = ∫_{B} ν_{F} d|∂F |_{𝔾}. (44)

Moreover, again from (42),

|∂E_{s_{j}}|_{𝔾}(U_{c}(0, R)) = \frac{1}{s_{j}^{Q−1}} |∂E|_{𝔾}(U_{c}(0, s_{j}R)) (45)

and, again for all ϕ ∈ C_{0}^{1}(𝔾, H𝔾),

∫_{U_{c}(0,R)} ⟨ϕ, ν_{E_{s_{j}}} ⟩ d|∂E_{s_{j}}|_{𝔾} = \frac{1}{s_{j}^{Q−1}} ∫_{U_{c}(0,s_{j}R)} ⟨ϕ ∘ δ_{\frac{1}{s_{j}}} , ν_{E}⟩ d|∂E|_{𝔾}

which yields, with a limiting procedure in ϕ,

∫_{U_{c}(0,R)} ν_{E_{s_{j}}} d|∂E_{s_{j}}|_{𝔾} = \frac{1}{s_{j}^{Q−1}} ∫_{U_{c}(0,s_{j}R)} ν_{E} d|∂E|_{𝔾}. (46)

Hence from (45) and (46) we have

∫_{U_{c}(0,R)} ν_{E_{s_{j}}} d|∂E_{s_{j}}|_{𝔾} = ∫_{U_{c}(0,s_{j}R)} ν_{E} d|∂E|_{𝔾}.

As j → +∞, the right hand side has limit ν_{E}(0), because 0 ∈ ∂_{𝔾}^{∗}E; hence

∫_{U_{c}(0,R)} ν_{E_{s_{j}}} d|∂E_{s_{j}}|_{𝔾} = (1 + o(1)) ν_{E}(0)|∂E_{s_{j}}|_{𝔾}(U_{c}(0, R))

and

∫_{U_{c}(0,R)} ⟨ν_{E}(0), ν_{E_{s_{j}}} ⟩ d|∂E_{s_{j}}|_{𝔾} = (1 + o(1)) |∂E_{s_{j}}|_{𝔾}(U_{c}(0, R)). (47)

Finally, by the semicontinuity of the perimeter,

|∂F |_{𝔾}(U_{c}(0, R)) ≤ lim inf_{j→+∞} |∂E_{s_{j}}|_{𝔾}(U_{c}(0, R))

and from (47)

= lim inf_{j→+∞} ∫_{U_{c}(0,R)} ⟨ν_{E}(0), ν_{E_{s_{j}}} ⟩_{ℝ^{m_{1}}} d|∂E_{s_{j}}|_{𝔾}

from (44) and for all R > 0 such that |∂F |_{𝔾}(∂U_{c}(0, R)) = 0,

= ∫_{U_{c}(0,R)} ⟨ν_{E}(0), ν_{F} ⟩_{ℝ^{m_{1}}} d|∂F |_{𝔾};

note that |∂F |_{𝔾}(∂U_{c}(0, R)) = 0 for L^{1}-a.e. R > 0. This concludes the proof of Claim 2 because |⟨ν_{E}(0), ν_{F} (x)⟩| ≤ 1, while |⟨ν_{E}(0), ν_{F} (x)⟩| = 1 precisely when ν_{F} (x) = ν_{E}(0).

Claim 3. There is q ∈ 𝔾 such that

F = τ_{q}(S_{𝔾}^{+}(ν_{E}(0)))

Proof of Claim 3. Take ζ ∈ ℝ^{m_{1}} and consider the left invariant differential operator

L_{ζ} := ∑_{j=1}^{m_{1}} ζ_{j}X_{j}

and the test function ϕ ∈ C_{0}^{1}(𝔾, H𝔾) whose canonical coordinates are ψ ζ with ψ ∈ C_{0}^{1}(𝔾, ℝ) arbitrary. Then

∫_{𝔾} 1_{F} L_{ζ}ψ dx = ∫_{𝔾} 1_{F} div_{𝔾} ϕ dx

= − ∫_{𝔾} ⟨ϕ, ν_{F} ⟩ d|∂F |_{𝔾} = − ∫_{𝔾} ψ⟨ζ, ν_{F} ⟩ d|∂F |_{𝔾}

Because ν_{F} (x) = ν_{E}(0) for all x, this yields that (in the sense of distributions)

\begin{cases}L_{ζ}1_{F} = 0 & if ⟨ζ, ν_{E}(0)⟩_{ℝ^{m_{1}}} = 0 \\ L_{ζ}1_{F} ≥ 0 & if ζ = ν_{E}(0). \end{cases} (48)

Thus, to achieve the proof of Claim 3, we are left to show that (48) implies: 1_{F} is constant along the integral curves of L_{ζ} when ζ is orthogonal to ν_{F}, while it is increasing along the integral curves of L_{ζ} when ζ = ν_{F}. This is the content of the following Lemma 3.2.

Note that Lemma 3.2 is the Carnot group version of the elementary Euclidean statement that \frac{∂f}{∂x_{2}} = ⋯ = \frac{∂f}{∂x_{n}} = 0 imply f = f(x_{1}). In Carnot groups the corresponding statement should be that the vanishing of X_{2}f to X_{m_{1}}f yields that f is a function of just one variable. But this is false as simple examples in the Heisenberg group ℍ^{1} show. What is possible to prove in step 2 groups is that if Y_{1}, …, Y_{m_{1}} are left invariant smooth orthonormal (horizontal) sections, Y_{2}f = ⋯ = Y_{m_{1}}f = 0 and Y_{1}f is positive then f is an increasing function of one variable. We notice here that for more complicated groups, as are groups of step 3 or larger, even this last weaker statement is false.

Lemma 3.2. Let 𝔾 be a step 2 group and let Y_{1}, …, Y_{m_{1}} be left invariant smooth orthonormal sections of H𝔾. Assume that g : 𝔾 → ℝ satisfies

Y_{1}g ≥ 0 and Y_{j}(g) = 0 if j = 2, …, m_{1}. (49)

Then the level lines of g are ”vertical hyperplanes orthogonal to Y_{1}” that is sets that are group translations of

S(Y_{1}) := {p : ⟨π_{0}p, Y_{1}(0)⟩ = 0}.

Proof. We prove the Lemma assuming that g is continuous and then we recover the general case approximating g by group convolution.

We prove first that

X_{j}g = 0 for j = m_{1} + 1, …, n. (50)

Indeed, suppose that (50) is proved. Recalling (see Proposition 2.2) that in a step 2 group X_{j} = \frac{∂}{∂x_{j}}, for j = m_{1} + 1, …, n, then (50) yields that g = g(x_{1}, …, x_{m_{1}}) and the thesis follows from (49) and an euclidean argument. We prove first (50) when 𝔾 is the three dimensional Heisenberg group ℍ^{1} ≡ ℝ^{3} whose points we indicate as [x, y, t]. The Lie algebra 𝔥 of ℍ^{1} is stratified as 𝔥 = 𝔥^{1} ⊕ 𝔥^{2}; 𝔥^{1} is generated by the left invariant vector fields X = \frac{∂}{∂x} + 2y \frac{∂}{∂t} and Y = \frac{∂}{∂y} − 2x\frac{∂}{∂t}; 𝔥^{2} is generated by T = −4\frac{∂}{∂t}. The commutation relations among these vector fields are: [X, Y] = T and all the other commutators vanish. In this case, rotations of ℝ^{3} leaving the t-axis fixed are both isometries and automorphisms of the group (see [64], Chapter XII, 7.5) and hence we can assume without loss of generality that Y_{1} = X. Our hypotheses then are

X ≡ Y_{1}(g) ≥ 0 and Y ≡ Y_{2}(g) = 0 (51)

and we want to prove that T (g) = 0. Observe first that, for all p, q ∈ ℍ^{1}, p = [p_{1}, p_{2}, p_{3}] and q = [q_{1}, q_{2}, q_{3}]

p_{1} > q_{1} ⟹ g(p) ≥ g(q). (52)

Indeed, to prove (52), it is enough to show that following the integral lines of Y with positive or negative direction and the integral lines of X only in the positive direction, – i.e. following a path along which g is either constant or increasing – it is possible to reach q from p. From the invariance of the vector fields X and Y under group translations it is enough to show that it is possible to connect the origin with any point [η, r_{2}, r_{3}] with η > 0. This fact can be verified directly and elementarily: first follow Y_{1} for a time s_{1} = r_{2}/2 + r_{3}/4η to reach [0, s_{1}, 0], then X_{1} for a time s_{2} = η > 0 to [η, s_{1}, r_{2}η + r_{3}/2] and, finally, Y_{1} once more for a time s_{3} = r_{2}/2−r_{3}/4η to reach [η, r_{2}, r_{3}]. This proves (52). Let us now define g^{+} : ℝ → ℝ as

g^{+}(s) := sup{g(p) : p = [x, y, t] ∈ ℍ, and x ≤ s}.

Clearly g^{+} is monotonically increasing and continuous given the continuity of g; moreover for all ε > 0

g^{+}(x − ε) ≤ g(p) ≤ g^{+}(x), ∀p = [x, y, t] ∈ ℍ (53)

hence g(p) = g^{+}(x) and T (g) = 0.

Let us consider now the general case when 𝔤 is generated by Y_{1}, …, Y_{m_{1}}, X_{m_{1}+1}, …, X_{n} and each X_{j} is a linear combination of commutators of the Y_{j}. Observe also that [Y_{i}, Y_{j}]g = 0 for all i, j = 2, …, m_{1}. Hence, to get (50), we have only to show that [Y_{1}, Y_{j}]g = 0 for j ∈ 2, …, m_{1}. Fix j_{0} ∈ 2, …, m_{1} and consider [Y_{1}, Y_{j_{0}}]. We assume that it does not vanish identically, otherwise there is nothing to prove. Let 𝔤̃ ⊂ 𝔤 be the subalgebra generated by Y_{1}, Y_{j_{0}} and [Y_{1}, Y_{j_{0}}]. Notice that 𝔤̃ is a subalgebra since 𝔤 has step 2, and that 𝔤̃ is isomorphic to 𝔥. Finally let 𝔾̃ := {exp v : v ∈ 𝔤̃} be the subgroup of 𝔾 with Lie algebra 𝔤̃. Clearly 𝔾̃ is isomorphic to ℍ^{1}. If g : 𝔾 → ℝ satisfies (49) then the restriction g_{|𝔾̃} of g to 𝔾̃ satisfies (51) hence [Y_{1}, Y_{j_{0}}](g) = 0 in 𝔾̃. We prove now that indeed [Y_{1}, Y_{j_{0}}]g(p) = 0, ∀p ∈ 𝔾. For any fixed p ∈ 𝔾, consider the function g̃ := g ∘ τ_{p}; by the invariance of the vector fields also Y_{1}g̃ ≥ 0 and Y_{j_{0}} g̃ = 0; hence by the same argument also [Y_{1}, Y_{j_{0}}]g̃(0) = 0. But observe that using once more the invariance of the vector fields

0 = [Y_{1}, Y_{j_{0}}]g̃(0) = [Y_{1}, Y_{j_{0}}](g ∘ τ_{p})(0) = [Y_{1}, Y_{j_{0}}]g(p).

This completes the proof.

Claim 4. The point q in Claim 3 can be chosen to be the origin, i.e.

F = S_{𝔾}^{+}(ν_{E}(0))

Proof of Claim 4. It is enough to prove that ⟨π_{0}q, ν_{E}(0)⟩_{0} = 0 because this yields that τ_{q}S_{𝔾}^{+}(ν_{E}(0)) = S_{𝔾}^{+}(ν_{E}(0)).

Indeed by (31), since 0 ∈ ∂_{𝔾}^{∗}E, there is k = k(𝔾) > 0 such that

(54)  lim inf_{r→0} min \{\frac{L^{n}(E ∩ U_{c}(0, r))}{L^{n}(U_{c}(0, r))} , \frac{L^{n}(E^{c} ∩ U_{c}(0, r))}{L^{n}(U_{c}(0, r))}\} = k.

Assume, by contradiction, that ⟨π_{0}q, ν_{E}(0)⟩_{0} > 0 then, if 0 < r̄ < ⟨π_{0}q, ν_{E}(0)⟩_{0}, clearly U_{c}(0, r̄) ∩ τ_{q}(S_{𝔾}^{+}(ν_{E}(0))) = ∅. On the contrary, from Claim 2, Claim 3 and (54) it follows

L^{n}(U_{c}(0, r̄) ∩ τ_{q}(S_{𝔾}^{+}(ν_{E}(0)))) = lim_{j→+∞} L^{n}(U_{c}(0, r̄) ∩ E_{s_{j}}) ≥ k L^{n}(U_{c}(0, r̄)) > 0.

If we assume ⟨π_{0}q, ν_{E}(0)⟩_{0} < 0 the contradiction is obtained working with E^{c}. This concludes the proof of Claim 4 that is of (38).

The second and last statement of the Blow-up Theorem is point (iii) of the following Lemma.

Lemma 3.3. Let p ∈ ∂_{𝔾}^{∗}E. Then

lim_{r→0} \frac{L^{n} (U_{c}(p, r) ∩ E ∩ τ_{p}S_{𝔾}^{−}(ν_{E}(p)))}{L^{n}(U_{c}(p, r))} = 0 (i)

lim_{r→0} \frac{L^{n} (U_{c}(p, r) ∩ E^{c} ∩ τ_{p}S_{𝔾}^{+}(ν_{E}(p)))}{L^{n}(U_{c}(p, r))} = 0 (ii)

lim_{r→0} \frac{|∂E|_{𝔾}(U_{c}(p, r))}{r^{Q−1}} = |∂S_{𝔾}^{+}(ν_{E}(p))|_{𝔾}(U_{c}(0, 1)). (iii)

Observe that the open balls U_{c}(p, r) can be replaced by the closed balls B_{c}(p, r).

In addition, (i), (ii), (iii) still hold when we replace cc-balls by d_{∞}-balls, if p belongs to the reduced boundary constructed with the distance d_{∞} (see Remark 2.28).

Proof. By translation, without loss of generality we may assume p = 0. Then

\begin{aligned}lim_{r→0} & \frac{L^{n}(U_{c}(0, r) ∩ E ∩ S_{𝔾}^{−}(ν_{E}(0)))}{L^{n}(U_{c}(0, r))} = lim_{r→0} \frac{1}{L^{n}(U_{c}(0, 1))} ∫_{𝔾} 1_{U_{c}(0,1)}1_{E_{r}}1_{S_{𝔾}^{−}(ν_{E}(0))} dL^{n} \\ & = \frac{1}{L^{n}(U_{c}(0, 1))} ∫_{𝔾} 1_{U_{c}(0,1)}1_{S_{𝔾}^{+}(ν_{E}(0))}1_{S_{𝔾}^{−}(ν_{E}(0))} dL^{n} = 0. \end{aligned}

This proves (i) and also (ii) replacing E with E^{c}. To prove (iii) observe that from the homogeneity of the perimeter we get

lim_{r→0} \frac{|∂E|_{𝔾}(U_{c}(0, r))}{r^{Q−1}} = lim_{r→0} |∂E_{r}|_{𝔾}(U_{c}(0, 1)).

Following the same computations as in Claim 2

ν_{E_{r}} |∂E_{r}|_{𝔾} ⇀ ν_{E}(0) |∂S_{𝔾}^{+}(ν_{E}(0))|_{𝔾} = ν_{E}(0) (H^{n−1} ∂S_{𝔾}^{+}). (55)

The last equality follows from Proposition 2.22. Now, by Proposition 2.4, (ii), with d = d_{c} and μ = |∂S_{𝔾}^{+}(ν_{E}(0))|_{𝔾} = H^{n−1}(∂S_{𝔾}^{+}(ν_{E}(0))), we have

|∂S_{𝔾}^{+}(ν_{E}(0))|_{𝔾}(∂U_{c}(0, 1)) = 0.

It follows that

lim_{r→0} ∫_{U_{c}(0,1)} ν_{E_{r}} d|∂E_{r}|_{𝔾} = ν_{E}(0) ∫_{U_{c}(0,1)} d|∂S_{𝔾}^{+}(ν_{E}(0))|_{𝔾}. (56)

Finally, since 0 ∈ ∂_{𝔾}^{∗}E, we have also that for r → 0

∫_{U_{c}(0,1)} ν_{E_{r}} d|∂E_{r}|_{𝔾} = (1 + o(1)) ν_{E}(0) |∂E_{r}|_{𝔾}(U_{c}(0, 1))

that together with (56) gives

\begin{aligned}lim_{r→0} |∂E_{r}|_{𝔾}(U_{c}(0, 1)) & = |∂S_{𝔾}^{+}(ν_{E}(0))|_{𝔾}(U_{c}(0, 1)) \\ & = H^{n−1}(∂S_{𝔾}^{+}(ν_{E}(0)) ∩ U_{c}(0, 1)). \end{aligned}

Assume now the point p belongs to the reduced boundary associated with the distance d_{∞} (as pointed out in Remark 2.28, this is true for |∂E|_{𝔾}-a.e. point p). If we replace now cc-balls by d_{∞}-balls, then (i) and (ii) follows straightforwardly from what we have just proved, since d_{c} and d_{∞} are equivalent doubling distances with respect to L^{n}. The last statement follows for d_{∞}-balls repeating our previous arguments, since, by Proposition 2.4, (ii), with d = d_{∞} and μ = |∂S_{𝔾}^{+}(ν_{E}(p))|_{𝔾} = H^{n−1}(∂S_{𝔾}^{+}(ν_{E}(p))), we have

|∂S_{𝔾}^{+}(ν_{E}(p))|_{𝔾}(∂U_{∞}(0, 1)) = 0.

Theorem 3.4. (Structure of 𝔾-Caccioppoli sets) If E ⊆ 𝔾 is a 𝔾-Caccioppoli set, then

(i) ∂_{𝔾}^{∗}E is (Q − 1)-dimensional 𝔾-rectifiable,

that is ∂_{𝔾}^{∗}E = N ∪ ⋃_{h=1}^{∞} K_{h}, where H_{c}^{Q−1}(N) = 0 and K_{h} is a compact subset of a 𝔾-regular hypersurface S_{h};

(ii) ν_{E}(p) is the 𝔾-normal to S_{h} at p, for all p ∈ K_{h};

|∂E|_{𝔾} = θ_{c}S_{c}^{Q−1} ∂_{𝔾}^{∗}E, (iii) where

θ_{c}(x) = \frac{1}{ω_{Q−1}} H^{n−1} (∂S_{𝔾}^{+}(ν_{E}(x)) ∩ U_{c}(0, 1)) .

As usual ω_{k} is the k-dimensional measure of the k-dimensional ball in ℝ^{k}. If we replace the S_{c}-measure by the S_{∞}-measure, the corresponding density θ_{∞} turns out to be a constant. More precisely

|∂E|_{𝔾} = θ_{∞} S_{∞}^{Q−1} ∂_{𝔾}^{∗}E, (iv)

where

θ_{∞} = \frac{ω_{m_{1}−1}ω_{m_{2}}ε_{2}^{m_{2}}}{ω_{Q−1}} = \frac{1}{ω_{Q−1}} H^{n−1} (∂S_{𝔾}^{+}(ν_{E}(0)) ∩ U_{∞}(0, 1)) .

Here ε_{2} is the constant appearing in (8).

Proof. Given our previous results, the proof follows closely a well known pattern as can be found e.g. in [3], [40], [25]. Hence we only sketch the argument following closely Theorem 2, Section 5.7.3 of [25].

By Lusin’s and Egoroff theorems, we can partition ∂_{𝔾}^{∗}E as

∂_{𝔾}^{∗}E = N ∪ ⋃_{h=1}^{∞} K_{h}.

Here |∂E|_{𝔾}(N) = 0 and each K_{h} is a compact set on which ν_{E} is continuous and such that the limits in Lemma 3.3 hold uniformly. Precisely, uniformly for q ∈ K_{h}, we have

\begin{aligned}\frac{L^{n} (U_{c}(q, r) ∩ E ∩ τ_{q}S_{𝔾}^{−}(ν_{E}(q)))}{L^{n}(U_{c}(q, r))} & → 0, as r → 0; \\ \frac{L^{n} (U_{c}(q, r) ∩ E^{c} ∩ τ_{q}S_{𝔾}^{+}(ν_{E}(q)))}{L^{n}(U_{c}(q, r))} & → 0, as r → 0. \end{aligned} (57)

Thus the first two parts of the theorem are proved by showing that each K_{h} is contained in a 𝔾-regular surface S_{h} whose 𝔾-normal coincides, on K_{h}, with ν_{E}. This is a consequence of Whitney extension theorem. Define

k_{c} = inf \{\frac{L^{n} (U_{c}(0, r) ∩ S_{𝔾}^{+}(v))}{L^{n} (U_{c}(0, r))} : v ∈ H𝔾_{0}\} .

By translation invariance, if p ∈ 𝔾, then

k_{c} = inf \{\frac{L^{n} (U_{c}(p, r) ∩ τ_{p}S_{𝔾}^{+}(v))}{L^{n} (U_{c}(p, r))} : v ∈ H𝔾_{p}\} .

Observe that if we use instead U_{∞} balls, that are invariant with respect to rotations of the horizontal space ℝ^{m_{1}}, the corresponding constant k_{∞} is k_{∞} = 1/2 (as it is in Euclidean spaces). Here we can only say that

0 < k_{c} ≤ 1/2.

The positivity of k_{c} is essential for our considerations and follows trivially from the comparability of d_{c} and d_{∞}. Then from (57), we get that for all ε > 0 there is \overline{r}(ε) > 0 such that for all r ∈ (0, \overline{r}(ε)), and for all q ∈ K_{h}

\begin{aligned}L^{n} (U_{c}(q, r) ∩ E ∩ τ_{q}S_{𝔾}^{−}(ν_{E}(q))) & < εL^{n} (U_{c}(q, r)) \\ L^{n} (U_{c}(q, r) ∩ E^{c} ∩ τ_{q}S_{𝔾}^{−}(ν_{E}(q))) & > (k_{c} − ε)L^{n} (U_{c}(q, r)) . \end{aligned} (58)

Claim. Define ϱ_{h} : ℝ^{+} → ℝ as

ϱ_{h}(t) := sup \{\frac{|⟨ν_{E}(p), π_{p}(p^{−1} · q)⟩_{p}|}{d_{c}(p, q)} : p, q ∈ K_{h}, 0 < d_{c}(p, q) < t\};

then

ϱ_{h}(t) → 0 as t → 0.

Proof. By contradiction, assume there is σ ∈ (0, 1) and a sequence {q_{i}}_{i} ⊂ K_{h}, q_{i} → p ∈ K_{h} as i → ∞, such that

⟨ν_{E}(p), π_{p}(p^{−1} · q)⟩_{p} > σd_{c}(p, q), (59)

or ⟨ν_{E}(p), π_{p}(p^{−1} · q)⟩_{p} < −σd_{c}(p, q). In both cases it is easy to see that

U_{c}(q_{i}, \frac{σ}{2} d_{c}(p, q_{i})) ⊂ U_{c}(p, 2d_{c}(p, q_{i})) ∩ τ_{p}S^{−}(ν_{E}(p)). (60)

Hence, from (58) and (60), if d_{c}(p, q_{i}) < \overline{r}(ε) it follows

εL^{n} (U_{c}(p, 2d_{c}(p, q_{i}))) > L^{n} (E ∩ U_{c}(p, 2d_{c}(p, q_{i})) ∩ τ_{p}S^{−}(ν_{E}(p)))

> L^{n} (E ∩ U_{c}(q_{i}, \frac{σ}{2} d_{c}(p, q_{i}))

> (k_{c} − ε)L^{n} (U_{c}(q_{i}, \frac{σ}{2} d_{c}(p, q_{i})) .

This gives a contradiction if ε is so small that

\frac{k_{c} − ε}{ε} > \frac{L^{n} (U_{c}(p, 2d_{c}(p, q_{i})))}{L^{n} (U_{c}(q_{i}, \frac{σ}{2} d_{c}(p, q_{i})))} = \frac{L^{n} (U_{c}(0, 2))}{L^{n} (U_{c}(0, \frac{σ}{2} ))}.

This completes the proof of our Claim.

Now apply Whitney extension theorem 2.14 with f = 0 and k = ν_{E}, to get the existence of a C_{𝔾}^{1} function f̃_{h} : 𝔾 → ℝ with f̃_{h} = 0 and |∇_{𝔾}f̃_{h}| ≠ 0 on K_{h}. We then define

S_{h} := {p ∈ 𝔾 : f̃_{h}(p) = 0 and |∇_{𝔾}f̃_{h}(p)| ≠ 0} .

This concludes the proof of (i) and (ii).

Let us prove (iii) and (iv).

¿From Lemma 2.26 we have

|∂E|_{𝔾} = |∂E|_{𝔾} ∂_{𝔾}^{∗}E. (61)

¿From Lemma 3.3 (iii), we get both

(62)  Θ_{c}^{∗ Q−1}(|∂E|_{𝔾}, p) := lim sup_{r→0} \frac{|∂E|_{𝔾}(U_{c}(p, r))}{ω_{Q−1}r^{Q−1}} = θ_{c},  for all p ∈ ∂_{𝔾}^{∗}E,

and

(63)  Θ_{∞}^{∗ Q−1}(|∂E|_{𝔾}, p) := lim sup_{r→0} \frac{|∂E|_{𝔾}(U_{∞}(p, r))}{ω_{Q−1}r^{Q−1}} = θ_{∞},  for all p ∈ ∂_{𝔾}^{∗}E.

¿From classical results of measure theory (see e.g. Theorems 2.10.17(2) and 2.10.19(3) in Federer’s book [26]), (62) and (63) imply (iii) and (iv).

The following divergence Theorem is then an easy consequence of Theorem 3.4, but we stress the fact that the measure theoretic boundary appears in the identity (ii). As in the Euclidean space, the corresponding statement for the reduced boundary holds straightforwardly. However, we prove here the statement for the measure theoretic boundary not only because – as in the Euclidean setting – the last one is sometimes easier to deal with, but mainly because the measure theoretic boundary – unlike the reduced boundary – is independent of the choice of the metric (see Remark 2.28).

Theorem 3.5. (Divergence Theorem) Let E be a 𝔾-Caccioppoli set, then

|∂E|_{𝔾} = θ_{∞} S_{∞}^{Q−1} ∂_{∗,𝔾}E, (i)

and the following version of the divergence theorem holds

− ∫_{E} div_{𝔾}ϕ dL^{n} = θ_{∞} ∫_{∂_{∗,𝔾}E}⟨ν_{E}, ϕ⟩ dS_{∞}^{Q−1}, ∀ϕ ∈ C_{0}^{1}(𝔾, H𝔾). (ii)

Proof. First observe that ∂_{𝔾}^{∗}E ⊂ ∂_{∗,𝔾}E, by (54), so that we need only to prove that

S_{c}^{Q−1}(∂_{∗,𝔾}E \ ∂_{𝔾}^{∗}E) = 0 (64)

To prove (ii) define for any ϵ > 0

A_{ϵ} := (𝔾 \ ∂_{𝔾}^{∗}E) ∩ {p ∈ 𝔾 : Θ_{c}^{∗ Q−1}(|∂E|_{𝔾}, p) > ϵ} .

Then, using once more Theorem 2.10.19(3) of [26], it follows that

|∂E|_{𝔾}(A_{ϵ}) ≥ ϵ S_{c}^{Q−1}(A_{ϵ}).

Because of Lemma 2.26 |∂E|_{𝔾}(𝔾 \ ∂_{𝔾}^{∗}E) = 0 and then

S_{c}^{Q−1}(A_{ϵ}) = 0 ∀ϵ > 0.

Then by an exhausting argument we get

Θ_{c}^{∗ Q−1}(|∂E|_{𝔾}, p) = 0 (65)

for S_{c}^{Q−1}-a.e. p ∈ 𝔾 \ ∂_{𝔾}^{∗}E.

To prove (i) thanks to (65), it is enough to prove that

Θ_{c}^{∗ Q−1}(|∂E|_{𝔾}, p) > 0 ∀p ∈ ∂_{∗,𝔾}E. (66)

¿From p ∈ ∂_{∗,E}, with elementary considerations, one gets the existence of δ > 0 and of a sequence {r_{i}}_{i}, r_{i} → 0 for i → +∞, such that

lim sup_{i→+∞} \frac{L^{n}(E ∩ U_{c}(p, r_{i}))}{L^{n}(U_{c}(p, r_{i}))} > δ  and  lim sup_{i→+∞} \frac{L^{n}(E^{c} ∩ U_{c}(p, r_{i}))}{L^{n}(U_{c}(p, r_{i}))} > δ.

Then the relative isoperimetric inequality (26) yields

δ^{\frac{Q−1}{Q}} L^{n}(U_{c}(p, r_{i}))^{\frac{Q−1}{Q}} ≤ c |∂E|_{𝔾}(U_{c}(p, r_{i})), ∀i ∈ ℕ

and the same for E^{c}. These two together imply (66).

An even sharper statement holds for Euclidean C^{1} domains. Indeed, extending a result proved by Z. Balogh in [7] for the Heisenberg group, in Section 4 we prove that the set C(S) of the characteristic points of an Euclidean C^{1} surface S ⊂ 𝔾 is such that H_{c}^{Q−1}(C(S)) = 0 and hence also S_{c}^{Q−1}(C(S)) = 0. Now, because non-characteristic points of a C^{1} boundary ∂E belong to the reduced boundary ∂_{𝔾}^{∗}E, it follows that S_{c}^{Q−1}(∂E \ ∂_{𝔾}^{∗}E) = 0. These facts and Proposition 2.22 prove the following

Corollary 3.6. If E has boundary of class C^{1} (and hence E is a 𝔾-Caccioppoli set), then

|∂E|_{𝔾} = θ_{∞} S_{∞}^{Q−1} ∂E = ( ∑_{j=1}^{m_{1}}⟨X_{j}, n⟩_{ℝ^{n}}^{2})^{1/2} H^{n−1} ∂E. (i)

where n_{E} denotes the Euclidean outward normal to ∂E. Again a version of the divergence theorem holds

− ∫_{E} div_{𝔾}ϕ dx = θ_{∞} ∫_{∂E}⟨ν_{E}, ϕ⟩ dS_{∞}^{Q−1} ∀ϕ ∈ C_{0}^{1}(𝔾, H𝔾). (ii)

4. Euclidean Regular and 𝔾-regular Hypersurfaces.

In this Section we study relationships between Euclidean notions of Hausdorff measure and of rectifiability, and the corresponding intrinsic notions in 𝔾. Our main result (see Theorem 4.10) states that in a step 2 group 𝔾 = ℝ^{n}, of homogeneous dimension Q ≥ n, Euclidean (n − 1)-dimensional rectifiable sets are (Q − 1)-dimensional 𝔾-rectifiable.

The opposite is false, even in the Heisenberg group ℍ^{1} = ℝ^{3}. Indeed, in [8] a set N ⊂ ℝ^{3} is constructed, such that for an appropriate ε > 0,

H_{c}^{3}(N) = 0 and H^{2+ε}(N) > 0.

Hence N is (trivially) 3 = (Q − 1)-dimensional ℍ^{1}-rectifiable, but it is not 2-dimensional Euclidean rectifiable because its Euclidean Hausdorff dimension is strictly larger than 2. We recall here that relationships between Euclidean and intrinsic Hausdorff measure in Heisenberg groups have been deeply investigated in [8], where also sharp results were obtained. We observe that for codimension 1 regular surfaces the situation is more complicated. Indeed an Euclidean C^{1} submanifold S is certainly 𝔾-rectifiable, as said before, but it is not necessarily 𝔾-regular, given the possible existence of characteristic points. On the contrary, an example proposed by Kirkhheim in a private communication, seems to show also that, not only fractal type sets as mentioned above, but also 𝔾-regular surfaces can have Euclidean dimension larger than n − 1.

A key step in the proof is provided by the construction of a suitable tiling of 𝔾, that in fact is not hierachic, and of a sharp estimate of the number of tiles of a given generation necessary to cover a tile of a different generation. Different kind of tilings appear in the literature, see for instance [17], [14], [61], [60]. Our tiling, in the special case of the Heisenberg group, was introduced, as far as we know, in [11].

Let C := {x ∈ ℝ^{n} : −1/2 < x_{i} ≤ 1/2 for i = 1, …, n}. We will call C the unit cube in 𝔾. For each λ > 0, C_{λ} is the dilated cube

C_{λ} = δ_{λ}C = {x : δ_{1/λ}x ∈ C} = {x ∈ ℝ^{n} : −λ^{α_{i}}/2 < x_{i} ≤ λ^{α_{i}}/2 for i = 1, …, n} .

Lemma 4.1. For each λ > 0 the family

P_{λ} = {τ_{p}C_{λ} : p ∈ δ_{λ}ℤ^{n}} (67)

is a partition of ℝ^{n}.

Proof. First we prove the theorem for λ = 1. It is enough to prove that for all y ∈ ℝ^{n} there are unique p ∈ Z^{n} and x ∈ C such that y = τ_{p}x. The first m_{1} (horizontal) coordinates of p and x are uniquely determined by the equations

p_{i} + x_{i} = y_{i} i = 1, …, m_{1}, p_{i} ∈ ℤ, −1/2 < x_{i} ≤ 1/2. (68)

For m_{1} + 1 ≤ i ≤ n the equations determining p_{i} and x_{i} are

(τ_{p}x)_{i} = p_{i} + x_{i} + Q_{i}(p, x) = y_{i} p_{i} ∈ ℤ, −1/2 < x_{i} ≤ 1/2 (69)

where (see (6)) for h_{j−1} < i ≤ h_{j}, Q_{i}(p, x) = Q_{i}(p_{1}, …, p_{h_{j−1}}, x_{1}, …, x_{h_{j−1}}). Thus, for m_{1} +1 ≤ i ≤ m_{1} + m_{2}, Q_{i} depends only on p_{1}, …, p_{m_{1}} and x_{1}, …, x_{m_{1}}, that we have already chosen according to (68). Hence (69) determines, in a unique way, p_{i} and x_{i} for m_{1} + 1 ≤ i ≤ m_{1} + m_{2}. The proof is completed by exhausting, step by step, all the layers.

To get the result for all λ > 0 observe that

δ_{λ}(τ_{p}C) = τ_{δ_{λ}p}C_{λ}. (70)

Hence

τ_{δ_{λ}p}C_{λ} ∩ τ_{δ_{λ}p′}C_{λ} = ∅ for p ≠ p′ ∈ ℤ^{n} (71) and

⋃_{p′∈δ_{λ}ℤ^{n}} τ_{p′}C_{λ} = ⋃_{p∈ℤ^{n}} τ_{δ_{λ}p}C_{λ} = ⋃_{p∈ℤ^{n}} δ_{λ}(τ_{p}C) = ℝ^{n}. (72)

Lemma 4.2. (Controlled Deformation) There is an increasing function d : ℝ^{+} → [1, +∞), depending only on the group 𝔾, such that

B_{c}(p, r) ⊂ B(p, d(||p||)r), ∀r ∈ (0, 1) and ∀p ∈ 𝔾, (73)

where || · || is the Euclidean norm in ℝ^{n} and B(p, r) is the Euclidean ball.

Proof. y ∈ B_{c}(p, r) if and only if y = p · z with z ∈ B_{c}(0, r). We have to estimate ||p · z − p||. Observe that, from Proposition 2.1, (p · z − p)_{i} = z_{i} + Q_{i}(p, z), for i = 1, …, n. Hence, again from Proposition 2.1,

|(p · z − p)_{i}| ≤ |z_{i}| + |Q_{i}(p, z)| ≤ c_{1}r + c_{2}(||p||)r = d(||p||)r.

Lemma 4.3. (Covering with metric balls) There is a positive decreasing function k : ℝ^{+} → (0, 1], depending only on the group 𝔾, such that any Euclidean ball B(p, r), r ∈ (0, 1), is contained, at most, in 20^{n}r^{n}/s^{Q} metric balls B_{c}(q_{i}, s), with s ≤ k(||p||)r.

Proof. First observe that if 2r < ||p − q|| ≤ 10r and if

s ≤ \frac{r}{d(||p|| + 10)} ≤ \frac{r}{d(||p|| + 10r)} ≤ \frac{r}{d(||q||)},

then, from Lemma 4.2 it follows that

B_{c}(q, s) ∩ B(p, r) = ∅. (74)

Then observe that the limitation ||p−q|| ≤ 10r can be dropped, indeed assume, by contradiction, the existence of q̃ ∈ 𝔾, 10r < ||p − q̃||, and s̃ ≤ \frac{r}{d(||p||+10)}, such that

B_{c}(q̃, s̃) ∩ B(p, r) ≠ ∅.

Then there is a geodesic line, for the cc-metric, from q̃ to a point of B(p, r) of length less than \frac{r}{d(||p||+10)}. This geodesic line crosses the shell \{q : 2r < ||p − q|| ≤ 10r\}; choose a point q′ of the geodesic that is also inside the shell; then, dist_{c}(q′, B(p, r)) < s̃ ≤ \frac{r}{d(||p||+10)}, so that B_{c}(q′, s̃) ∩ B(p, r) ≠ ∅ in contradiction with (74). Hence we have proved that

B_{c}(q, s) ∩ B(p, r) = ∅, if s ≤ \frac{r}{d(||p|| + 10)} and 2r < ||q − p||. (75)

¿From the comparability of d_{∞} and of d_{c}, there is c_{1} ≥ 1 such that B_{c}(q, c_{1}λ) ⊃ τ_{q}C_{λ}, for all q ∈ 𝔾 and λ > 0. Since the family P_{λ} defined in (67) is a partition of ℝ^{n} then the family

F_{s} := {B_{c}(q, c_{1}s) : q ∈ δ_{s}ℤ^{n}}

is a covering of 𝔾.

¿From (75), if s ≤ rk(||p||) := \frac{r}{c_{1}d(||p||+10)}, the subfamily of F_{s} of the metric balls that intersect (and obviously cover) B(p, r) contains only balls with center inside B(p, 10r). It is then easy to check that the cardinality of the set of these centers is less than \frac{20^{n}r^{n}}{s^{Q}}.

The following proposition gives a comparison among Euclidean Hausdorff measure and intrinsic Hausdorff measure.

28 BRUNO FRANCHI, RAUL SERAPIONI, AND FRANCESCO SERRA CASSANO

Proposition 4.4. Let 𝔾 be a Carnot group. For any α ≥ 0 and R > 0 there is a constant c(α, R) > 0 such that for any set E ⊂ 𝔾 ∩ B(0, R) H_{c}^{α+Q−n}(E) ≤ c(α, R)H^{α}(E), α ≥ 0. (76) In particular, for all E ⊂ 𝔾, H^{α}(E) = 0 ⟹ H_{c}^{α+Q−n}(E) = 0, α ≥ 0. (77)

Proof. Assume H^{α}(E) < ∞ and E ⊂ B(0, R). Then for any δ > 0 there is a covering of E consisting of Euclidean balls B(p_{i}, r_{i}), i = 1, …, N, and with radius less than δ such that

∑_{i=1}^{N} r_{i}^{α} ≤ 2H^{α}(E) < ∞.

From Lemma 4.3 each Euclidean ball B(p_{i}, r_{i}) can be covered by a precise number of metric balls B_{c}(q_{ij}, s_{i}) with radius s_{i} comparable with r_{i}, precisely

B(p_{i}, r_{i}) ⊂ ⋃_{j=1}^{N_{i}} B_{c}(q_{ij}, s_{i}),

where s_{i} = k(2R)r_{i} and N_{i} ≤ 20^{n}r_{i}^{n}/s_{i}^{Q} = 20^{n}k(2R)^{−Q}r_{i}^{n−Q}. Hence

∑_{i=1}^{N} ∑_{j=1}^{N_{i}} s_{i}^{Q−n+α} ≤ ∑_{i=1}^{N} 20^{n}k(2R)^{−Q}r_{i}^{n−Q} (k(2R)r_{i})^{Q−n+α} ≤ 20^{n}k(2R)^{α−n} ∑_{i=1}^{N} r_{i}^{α}.

This completes the proof of (76). To get (77) simply consider E as the union of E ∩ B(0, R) with R → ∞.

For each fixed integer i_{0}, let

Π_{i_{0}} : ℝ^{n} → {x : x_{i_{0}} = 0}

be the Euclidean orthogonal projection: x ↦ Π_{i_{0}}x := (x_{1}, …, x_{i_{0}−1}, 0, x_{i_{0}+1}, …, x_{n}). Lemma 4.5. If 𝔾 is a Carnot group and i_{0} belongs to the higher layer of 𝔤, that is m_{1} + ⋯ + m_{k−1} + 1 ≤ i_{0} ≤ n, or, in particular, if 𝔾 is a step 2 group and m_{1} + 1 ≤ i_{0} ≤ n, then the family P′_{λ} = {Π_{i_{0}}τ_{p}C_{λ} : p ∈ δ_{λ}ℤ^{n}} (78) is a partition of the plane {x : x_{i_{0}} = 0}.

Proof. It is enough to observe that Π_{i_{0}}τ_{p}C_{λ} ∩ Π_{i_{0}}τ_{p′}C_{λ} ≠ ∅ ⟹ Π_{i_{0}}τ_{p}C_{λ} = Π_{i_{0}}τ_{p′}C_{λ}. (79) Indeed let

(80) y ∈ τ_{p}C_{λ}, y′ ∈ τ_{p′}C_{λ} be such that Π_{i_{0}}y = Π_{i_{0}}y′. If y = p · x and y′ = p′ · x′, with x, x′ ∈ C_{λ}, then

x_{i} + p_{i} = x′_{i} + p′_{i} for 1 ≤ i ≤ m_{1}

so that, because |x_{i} − x′_{i}| < λ and (p_{i} − p′_{i})/λ ∈ ℤ, it follows x_{i} = x′_{i} and p_{i} = p′_{i}, for 1 ≤ i ≤ m_{1}. (81) Consider now the vertical components different from i_{0}, for them we have once more

x_{i} + p_{i} + Q_{i}(p, x) = x′_{i} + p′_{i} + Q_{i}(p′, x′) for m_{1} + 1 ≤ i ≤ n, i ≠ i_{0}.

Assume m_{1} + 1 ≤ i ≤ m_{1} + m_{2}, then from (81) and (6) we have that Q_{i}(p, x) = Q_{i}(p′, x′), so that it follows again that x_{i} = x′_{i} and p_{i} = p′_{i} also for m_{1} + 1 ≤ i ≤ m_{1} + m_{2}. Iterate this procedure, exhausting step by step all the vertical layers, to get eventually

x_{i} = x′_{i} and p_{i} = p′_{i}, for m_{1} + 1 ≤ i ≤ n, i ≠ i_{0}.

This way we have proved that only the i_{0} components of p and p′ may be different, hence (p · x)_{i} = (p′ · x)_{i}, for i ≠ i_{0}, so that

Π_{i_{0}}τ_{p}C_{λ} = Π_{i_{0}}τ_{p′}C_{λ}.

The elements of P′_{λ} clearly are not Euclidean parallelepypeds (but in the case in which i_{0} is the only vertical direction as in the Heisenberg groups). We are not interested in the actual shape of these sets, but we are interested in estimating how many elements of P′_{λ} are necessary to cover the intersection of a given box C_{δ} with the plane {x : x_{i_{0}} = 0} when λ is much smaller than δ. This is the content of the next Lemma. Notice that the following Lemmas, unlike the previous ones of this section, hold only for step 2 groups.

Lemma 4.6. Let 𝔾 be a step 2 group and m_{1} + 1 ≤ i_{0} ≤ n. For a given positive integer M let U = C_{1/M} ∩ {x : x_{i_{0}} = 0}. Then for any integer N > M the set U can be covered by

c [\frac{N}{M} ]^{m_{1}+2(n−m_{1}−1)} elements of the family P′_{1/N}, where c = c(𝔾) is a positive constant depending only on the group 𝔾.

Proof. We will prove that there is a constant c′ = c′(𝔾) ≥ 1 such that for any y ∈ U it is possible to find a (unique) p = (p_{1}, …, p_{n}) ∈ δ_{1/N}ℤ^{n} such that y ∈ Π_{i_{0}}τ_{p}C_{1/N}, with p satisfying also

max_{i=1,...,m_{1}} |p_{i}| ≤ \frac{1}{M} max_{i=m_{1}+1,...,n} |p_{i}| ≤ \frac{c′}{M^{2}} , p_{i_{0}} = 0.

The thesis then follows with c = (c′ + 1)^{m_{1}+2(n−m_{1}−1)}.

Given y ∈ U, to prove that y ∈ Π_{i_{0}}τ_{p}C_{1/N} is equivalent to find x ∈ C_{1/N} and p ∈ δ_{1/N}ℤ^{n} such that

y_{i} = p_{i} + x_{i} ≡ \frac{h_{i}}{N} + x_{i} where h_{i} ∈ ℤ, −\frac{1}{2N} < x_{i} ≤ \frac{1}{2N} , when 1 ≤ i ≤ m_{1}

y_{i} = p_{i} + x_{i} + Q_{i}(p, x) ≡ \frac{h_{i}}{N^{2}} + x_{i} + Q_{i}(p, x),

where h_{i} ∈ ℤ, −\frac{1}{2N^{2}} < x_{i} ≤ \frac{1}{2N^{2}} , when m_{1} + 1 ≤ i ≤ n, i ≠ i_{0},

y_{i_{0}} = 0.

If y is given, the first m_{1} equations can be solved uniquely with |h_{i}| ≤ \frac{N}{M} that implies |p_{i}| ≤ \frac{1}{M}.

To solve the second group of n − m_{1} − 1 equations, remember once more that Q_{i} is a homogeneous polynomial of degree 2, depending only on the first m_{1} components of p and x. Hence if x ∈ C_{1/N} and p is determined as before then

|Q_{i}(P, x)| ≤ c′′ max_{i=1,...,m_{1}} |x_{i}| max_{i=1,...,m_{1}} |p_{i}| ≤ c′′ \frac{1}{M^{2}} , (82)

where c′′ = c′′(𝔾). If y ∈ U, |y_{i}| ≤ \frac{1}{2M^{2}} for m_{1} + 1 ≤ i ≤ n, i ≠ i_{0}, then also each equation of the second group has a unique solution p_{i}, x_{i} with |x_{i}| ≤ \frac{1}{2N^{2}} and |p_{i}| ≤ c′ \frac{1}{M^{2}} . This completes the proof of the Lemma.

The following lemma gives a quantitative estimate of the fact that an Euclidean C^{1} surface near a characteristic point p is close to the horizontal fiber through p.

Lemma 4.7. As in the previous Lemma, let 𝔾 be a step 2 group, m_{1} + 1 ≤ i_{0} ≤ n. If N > M > 0, write U = C_{1/M} ∩ \{x : x_{i_{0}} = 0\} and V = C_{1/N} ∩ \{x : x_{i_{0}} = 0\}. Assume that ϕ : \overline{U} → ℝ is of Euclidean class C^{1} in \overline{U} and that

ϕ(0) = 0 and \frac{∂ϕ}{∂x_{j}} (0) = 0, for j = 1, …, m_{1}. (83)

Then for any ε > 0 it is possible to choose N = N(ε, 𝔾, ϕ) > M such that the surface Σ, the graph of ϕ over V, i.e.

Σ := {x ∈ 𝔾 : x′ := Π_{i_{0}}x ∈ V and x_{i_{0}} = ϕ(x′)}

is contained in the union of, at most, [εN] translates of the cube C_{1/N}.

Proof. From X_{j}ϕ(0) ≡ \frac{∂ϕ}{∂x_{j}} (0) = 0 for j = 1, …, m_{1} and the uniform continuity of the partial

derivatives of ϕ in U, it follows that for any ε > 0 it is possible to choose N > M such that ∀x′ ∈ V

|\frac{∂ϕ}{∂x_{j}} (x′)| < ε, for j = 1, …, m_{1},

while for all the other derivatives we have

|\frac{∂ϕ}{∂x_{j}} (x′)| ≤ sup_{x′∈U} |\frac{∂ϕ}{∂x_{j}} (x′)| = c < ∞, for j = m_{1} + 1, …, n, j ≠ i_{0}.

By definition, if x′ ∈ V then

|x′_{j}| ≤ \frac{1}{2N} , for 1 ≤ j ≤ m_{1} and |x′_{j}| ≤ \frac{1}{2N^{2}} , for m_{1} + 1 ≤ j ≤ n, j ≠ i_{0},

hence

|ϕ(x′)| ≤ n max\{\frac{ε}{2N} , \frac{c}{2N^{2}} \}, ∀x′ ∈ V.

Given ε > 0 we can choose N = N(ε, 𝔾, ϕ) so that |ϕ(x′)| < \frac{ε}{2N} ∀x′ ∈ V. Hence the graph Σ of ϕ is contained inside the box B = \{x ∈ 𝔾 : x′ ∈ V, |x_{i_{0}}| < ε/2N\}. Finally, let q^{k} ∈ \frac{1}{N} ℤ^{n} ⊂ 𝔾 be points such that only their vertical components (q^{k})_{i_{0}} := \frac{k}{N^{2}} do not vanish. Then from (4),(5),(6) it follows that

B ⊂ ⋃_{k=−[(εN)/2]}^{[(εN)/2]} τ_{q^{k}}C_{\frac{1}{N}} ,

i.e. B is contained in a vertical pile of [εN] translated cubes C_{1/N} in the direction i_{0}. We observe that the hypothesis that 𝔾 is a step 2 group has been used only at this last point.

Our main theorem describes the size of the characteristic set C(S) of a regular surface S in terms of its intrinsic Hausdorff dimension. This result refines previous related results in [24], [35], [16]. The proof relies on a technique developed by Z.Balogh in [7] for the Heisenberg group.

Theorem 4.8. If S is a Euclidean C^{1}-smooth hypersurface in a step 2 group 𝔾 with homogeneous dimension Q = m_{1} + 2(n − m_{1}), then

H_{𝔾}^{Q−1}(C(S)) = 0. (84)

Proof. Our proof follows the approach in [7]. We will prove that for any p ∈ C(S) there is a positive number M, such that

H_{𝔾}^{Q−1}(C(S) ∩ τ_{p}C_{1/M}) = 0. (85)

Since C(S) is a closed set the thesis will follow by a compactness argument. Moreover it is possible to assume p = 0 in (85), because the characteristic set of the translated surface τ_{−p}S ∩ C_{1/M} is just τ_{−p}C(S) ∩ C_{1/M} and the intrinsic Hausdorff measure H_{𝔾}^{Q−1} is invariant under translations of the group. Hence it is enough to prove that if 0 ∈ C(S) there is M > 0 such that

H_{𝔾}^{Q−1}(C(S) ∩ C_{1/M}) = 0. (86)

To this aim we recall the definition of s-dimensional Hausdorff content H_{𝔾}^{s,∞} of a subset A ⊂ 𝔾

H_{𝔾}^{s,∞}(A) = inf{∑_{j}(diam_{𝔾}A_{j})^{s} : A ⊆ ∪_{j}A_{j}} (87)

It is clear that the inequality H_{𝔾}^{s,∞}(A) ≤ H_{𝔾}^{s}(A) always holds, but it is also true that H_{𝔾}^{s,∞}(A) = 0 implies H_{𝔾}^{s}(A) = 0 (see [50]). Hence to get (86) it is enough to prove that for any ε > 0

H_{𝔾}^{Q−1,∞}(C(S) ∩ C_{1/M}) < ε. (88)

By definition there are M > 0 and an Euclidean C^{1} function f : C_{2/M} → ℝ such that S ∩ C_{2/M} = {x ∈ 𝔾 : f(x) = 0} with |∇f(x)| ≠ 0 in C_{2/M} and |∇_{𝔾}f(0)| = 0 because 0 ∈ C(S). Hence ∇f(0) = ∑_{j=m_{1}+1}^{n} λ_{j}X_{j}(0) = ∑_{j=m_{1}+1}^{n} λ_{j}e_{j} for appropriate λ_{j} ∈ ℝ and there is at least one of the λ_{j} different from 0. Assume that λ_{i_{0}} ≠ 0, i.e.

X_{i_{0}}f(0) ≡ \frac{∂f}{∂x_{i_{0}}} (0) = λ_{i_{0}} ≠ 0.

By the Euclidean implicit function theorem, if M is small enough, S ∩ C_{2/M} is the graph, in the direction of X_{i_{0}}(0) = e_{i_{0}}, of a C^{1} function ψ of n − 1 variables, ψ : \overline{2U} → ℝ where 2U = C_{2/M} ∩ {x : x_{i_{0}} = 0} that is

S ∩ C_{2/M} = {x ∈ 𝔾 : x′ := Π_{i_{0}}x ∈ 2U and x_{i_{0}} = ψ(x′)}.

Consider, as in Lemmas 4.5 and 4.6, the partition P′_{1/2N} of U = C_{1/M} ∩ {x : x_{i_{0}} = 0} where N > M will be chosen later appropriately large. Let us denote by C^{h} the elements of P′_{1/2N} and let S^{h} be the piece of S that is graph over C^{h} ∩U. Consider only the sets S^{h} that intercept C(S). If S^{h} is one of these, choose one q^{h} ∈ C(S) ∩ S^{h} and translate S^{h} so that q_{h} goes to the origin. It is possible to see that also τ_{−q_{h}}S^{h} is a graph in direction i_{0} of an Euclidean C^{1} function ϕ. Finally we define Σ as the graph of ϕ (in the direction i_{0}), that is

Σ = {x ∈ 𝔾 : x′ := Π_{i_{0}}x ∈ Π_{i_{0}}C_{1/N} and x_{i_{0}} = ϕ(x′)}

It is immediate to verify that τ_{−q^{h}}S^{h} ⊂ Σ. Hence by Lemma 4.7 we get that for any ε > 0 it is possible to find N > 0 such that Σ and consequently τ_{−q^{h}}S^{h} are contained in a pile of at most [εN] translated cubes as C_{1/N} so that, in turn, also S^{h} is contained in the union of at most [εN] sets that are translations of the original cube C_{1/N}.

Since, by Lemma 4.6, U is covered by k [\frac{N}{M} ]^{m_{1}+2(n−m_{1}−1)} elements of the family P′_{1/2N}

it follows that C(S) ∩ C_{1/M} is contained in at most kεN(\frac{N}{M})^{m_{1}+2(n−m_{1}−1)} = c(𝔾, M)εN^{Q−1} translated of the cube C_{1/N}. Because

diam_{𝔾}(τ_{p}C_{1/N}) = diam_{𝔾}C_{1/N} ≤ c(𝔾)\frac{1}{N}

by definition of H_{𝔾}^{Q−1,∞} content, given in (87), we have

H_{𝔾}^{m_{1}+2(n−m_{1})−1,∞}(C(S) ∩ C_{1/M}) ≤ c(𝔾, M)ε. (89)

that is (88), and the proof is concluded.

Corollary 4.9. Let 𝔾 = ℝ^{n} be a step 2 Carnot group. If S is an Euclidean C^{1}-smooth hypersurface of ℝ^{n} then S is (Q − 1)-dimensional 𝔾-rectifiable.

Proof. From the previous Theorem 4.8,

S = (S \ C(S)) ∩ C(S), with H_{c}^{Q−1}(C(S)) = 0,

then the thesis follows because S \ C(S) is 𝔾-regular.

Theorem 4.10. Let 𝔾 = ℝ^{n} be a step 2 Carnot group. Then, if S is a (n − 1)-dimensional Euclidean rectifiable set of ℝ^{n} then S is also (Q − 1)-dimensional 𝔾-rectifiable.

Proof. By definition S = ⋃_{h=1}^{∞} K_{h} ∩ N, with K_{h} ⊂ S_{h}, S_{h} Euclidean C^{1}-smooth and H^{n−1}(N) = 0. The previous Corollary 4.9 yields that S_{h}, and hence K_{h}, are 𝔾-rectifiable. Proposition 4.4 implies H_{c}^{Q−1}(N) = 0, so that also N is 𝔾-rectifiable. The thesis follows because a countable union of rectifiable sets is rectifiable.

5. Appendix.

Theorem 5.1. There exist positive constants ε_{1}, …, ε_{k} ∈ (0, 1) depending only on the group structure such that, if we put

d_{∞}(x, y) = d_{∞}(y^{−1} · x, 0),

where, if p = [p^{1}, …, p^{k}] ∈ ℝ^{m_{1}} × ⋯ × ℝ^{m_{k}} = ℝ^{n}, and

d_{∞}(p, 0) = max{ε_{j}(|p^{j}|_{ℝ^{m_{j}}})^{1/j} , j = 1, …, k},

then d_{∞} is a distance in 𝔾.

Proof. The choice of ε_{1}, …, ε_{k} is done by iteration: choose first ε_{1} = 1; then suppose ε_{1}, …, ε_{i−1} have been chosen. The choice of ε_{i} can be done as follows: by (5) and (6), if x = (x_{1}, …, x_{n}) = [x^{1}, …, x^{k}], y = (y_{1}, …, y_{n}) = [y^{1}, …, y^{k}] ∈ 𝔾, and h_{i−1} < ℓ ≤ h_{i} then the function Q_{ℓ}(x, y) appearing in the group law (4) can be written as a sum of terms of the form

q_{ℓ;θ_{1},...,θ_{h_{i−1}},σ_{1},...,σ_{h_{i−1}}} (x_{1})^{θ_{1}} ⋯ (x_{h_{i−1}})^{θ_{h_{i−1}}} (y_{1})^{σ_{1}} ⋯ (y_{h_{i−1}})^{σ_{h_{i−1}}} , (90)

where q_{ℓ;θ_{1},...,θ_{h_{i−1}},σ_{1},...,σ_{h_{i−1}}} ∈ ℝ, and the sum is extended to all non negative integers θ_{1}, …, θ_{h_{i−1}}, σ_{1}, …, σ_{h_{i−1}} such that

∑_{j=1}^{h_{i−1}} θ_{j} > 0 , ∑_{j=1}^{h_{i−1}} σ_{j} > 0 (91)

and

∑_{j=1}^{h_{i−1}} α_{j}θ_{j} + ∑_{j=1}^{h_{i−1}} α_{j}σ_{j} = α_{ℓ} = i. (92)

(indeed α_{ℓ} = i, by (3)). Since, by definition, if h_{j−1} < λ ≤ h_{j} then

|x_{λ}| ≤ \frac{1}{ε_{j}^{j}} d_{∞}(x, 0)^{j} , |y_{λ}| ≤ \frac{1}{ε_{j}^{j}} d_{∞}(y, 0)^{j}, (93)

it follows from (92) that |Q_{ℓ}(x, y)| is bounded by

\begin{aligned}∑_{ν=1}^{i−1} ( ∑_{∑{h_{i−1} \atop j=1} α_{j} θ_{j} =ν ; ∑{h_{i−1} \atop j=1} α_{j} σ_{j} =i−ν} |q_{ℓ;θ_{1},...,θ_{h_{i−1}},σ_{1},...,σ_{h_{i−1}}} | & ε_{1}^{− ∑_{j=1}^{h_{1}}(θ_{j}+σ_{j})} ⋯ ε_{h_{i−1}}^{−(i−1) ∑_{j=h_{i−2}+1}^{h_{i−1}}(θ_{j}+σ_{j})}) \\ & · d_{∞}(x, 0)^{ν}d_{∞}(y, 0)^{i−ν} \end{aligned}

(by (91), the terms corresponding to ν = 0 or ν = i do not appear in the expression of Q_{i}). We choose now ε_{i} such that

ε_{i}^{i} ∑_{∑{h_{i−1} \atop j=1} α_{j} θ_{j} =ν ; ∑{h_{i−1} \atop j=1} α_{j} σ_{j} =i−ν} |q_{ℓ;θ_{1},...,θ_{h_{i−1}},σ_{1},...,σ_{h_{i−1}}} | ≤ \binom{ν}{i} ε_{1}^{∑_{j=1}^{h_{1}}(θ_{j}+σ_{j})} ⋯ ε_{i−1}^{(i−1) ∑_{j=h_{i−2}+1}^{h_{i−1}}(θ_{j}+σ_{j})}, (94)

for ν = 1, …, i − 1, so that

ε_{i}^{i} |Q_{ℓ}(x, y)| ≤ ∑_{ν=1}^{i−1} \binom{ν}{i} d_{∞}(x, 0)^{ν}d_{∞}(y, 0)^{i−ν} (95)

for h_{i−1} < ℓ ≤ h_{i}, i = 1, …, k.

To achieve the proof of the theorem, let us prove that, if ε_{1}, …, ε_{k} are chosen according (94), then d_{∞} is a distance in 𝔾. Clearly, to this end we have only to prove that

d_{∞}(x · y, 0) ≤ d_{∞}(x, 0) + d_{∞}(y, 0). (96)

Now, to prove (96), put p = x · y, and let i ∈ {1, …, k} be such that

d_{∞}(x · y, 0) = ε_{i}(|p^{i}|_{ℝ^{m_{i}}})^{1/i}.

Then, by (93) and (95),

d_{∞}(x · y, 0)^{2i} ≤ ∑_{ℓ=h_{i−1}+1}^{h_{i}} (ε_{i}^{i}|x_{ℓ}| + ε_{i}^{i}|y_{ℓ}| + ε_{i}^{i}|Q_{ℓ}(x, y)|)^{2}

≤ (d_{∞}(x, 0)^{i} + d_{∞}(y, 0)^{i} + ∑_{ν=1}^{i−1} \binom{ν}{i} d_{∞}(x, 0)^{ν}d_{∞}(y, 0)^{i−ν})^{2}

= (d_{∞}(x, 0) + d_{∞}(y, 0))^{2i} ,

and the theorem is completely proved.

Theorem 5.2. (Whitney Extention Theorem) Let F ⊂ 𝔾 be a closed set, and let f : F → ℝ, k : F → H𝔾 be, respectively, a continuous real function and a continuous horizontal section. We set

R(x, y) := \frac{f(x) − f(y) − ⟨k(y), π_{y}(y^{−1} · x)⟩_{y}}{d_{c}(y, x)} ,

and, if K ⊂ F is a compact set,

ρ_{K}(δ) := sup{|R(y, x)| : y, x ∈ K, 0 < d_{c}(y, x) < δ}.

Assume

((i)) ρ_{K}(δ) → 0 as δ → 0 for every compact set K ⊂ F,

then there exist f̃: 𝔾 → ℝ, f̃ ∈ C_{𝔾}^{1}(𝔾) such that

f̃_{|F} ≡ f, ∇_{𝔾}f̃_{|F} ≡ k.

Proof. The proof follows closely the one in Euclidean spaces as can be found e.g. in Section 6.5 of [25].

Let θ > 1 be a fixed constant to be chosen later. If p ∈ 𝔾 let r(p) := \frac{1}{20θ} min\{1, dist_{c}(p, F)\} where, clearly, dist_{c}(p, F) := inf{d_{c}(p, q) : q ∈ F }. Vitali covering theorem yields the existence of a countable set S ⊂ 𝔾 \ F such that

𝔾 \ F = ⋃_{p∈S} U_{c}(p, 5r(p))

and such that the balls U_{c}(p, r(p)) are two by two disjointed. Fix q ∈ 𝔾 \ F and define

S_{q} := {p ∈ S : U_{c}(q, 10θr(q)) ∩ U_{c}(p, 10θr(p)) ≠ ∅}.

First observe that if p ∈ S_{q}, then

\frac{1}{3} ≤ \frac{r(q)}{r(p)} ≤ 3, ∀p ∈ S_{q}; (97)

indeed, if dist_{c}(p, F), dist_{c}(q, F) ≤ 1, then

20θr(q) ≤ dist_{c}(q, F)
≤ d_{c}(p, q) + dist_{c}(p, F)
≤ 10θ(r(p) + r(q)) + dist_{c}(p, F) ≤ 10θ(r(p) + r(q)) + 20θr(p)

that gives r(q) ≤ 3r(p). Changing the role of p and q we get (97). As a consequence of (97) it follows that for all q ∈ 𝔾 \ F,

Card(S_{q}) ≤ (120θ + 9)^{Q}. (98)

Indeed, if p ∈ S_{q} then p ∈ U_{c}(q, 40θr(q)) and hence

U_{c}(p, r(p)) ⊂ U_{c}(q, 40θr(q) + r(p)) ⊂ U_{c}(q, (40θ + 3)r(q)).

Now, recalling that the balls U_{c}(p, r(p)) for p ∈ S are disjointed and that their volume is estimated as in (11), we get

Card(S_{q})L^{n}(U_{c}(0, 1))(\frac{r(q)}{3})^{Q} ≤ ∑_{p∈S_{q}} L^{n}(U_{c}(p, r(p)))
≤ L^{n}(U_{c}(q, (40θ + 3)r(q)))
= L^{n}(U_{c}(0, 1))(40θ + 3)^{Q}r(q)^{Q},

and (98) is proved.

Choose now δ̂: 𝔾 → ℝ as

δ̂(p) := (∑_{i=1}^{n} |p_{i}|^{2 \frac{α_{1}...αn}{α_{i}}} )^{\frac{1}{2α_{1}...αn}} ,

where α_{i} is the homogeneity of the variable p_{i} as defined in (3). Then define δ : 𝔾 × 𝔾 → ℝ as δ(p, q) := δ̂(q^{−1} · p). Observe that δ̂ ∈ C^{∞}(𝔾 \ {0}), that it is homogeneous of degree 1 with respect to the dilations of 𝔾 and consequently, δ is comparable with d_{c}(x, 0), that is

c_{1}δ(p, q) ≤ d_{c}(p, q) ≤ c_{2}δ(p, q), for p, q ∈ 𝔾,

where, respectively, c_{1} := inf{d_{c}(x, 0) : δ̂(x) = 1} > 0 and c_{2} := sup{d_{c}(x, 0) : δ̂(x) = 1} < ∞. Fix now the constant θ in (98) as θ := \frac{c_{2}}{c_{1}} > 1.

Now let μ : ℝ → ℝ be a smooth non increasing function such that

μ(t) := \{ \begin{matrix}1 & for & t ≤ 1/c_{1} \\ 0 & for & t ≥ 2/c_{1}, \end{matrix}

and, for any p ∈ S, define g_{p} : 𝔾 → ℝ as

g_{p}(x) := μ (\frac{δ(p, x)}{5r(p)}) . (99)

The following properties of g_{p} are easy to check: g_{p} ∈ C^{∞}(𝔾); 0 ≤ g_{p}(x) ≤ 1;

g_{p}(x) = \{ \begin{matrix}1 & for & x ∈ U_{c}(p, 5r(p)) \\ 0 & for & x ∈ 𝔾 \ U_{c}(p, 10θr(p)); \end{matrix}

there are uniform bounds on X_{j}g_{p}, that is there are positive constants c_{3} and c_{4} such that

(100)  |X_{j}g_{p}(x)| ≤ \frac{c_{3}}{r(p)} ≤ \frac{c_{4}}{r(x)},  for j = 1, …, m_{1} and for all x ∈ 𝔾.

The last inequality follows from the fact that |X_{j}g_{p}(x)| ≠ 0 only if x ∈ U_{c}(p, 10θr(p)) \ U_{c}(p, 5r(p)) and in this set r(p) is comparable with r(x).

Now define σ : 𝔾 → ℝ as

σ(x) := ∑_{p∈S} g_{p}(x).

Because g_{p} = 0 on U_{c}(q, 10θr(q)) if p ∉ S_{q}, then the above sum is locally finite, precisely

σ = ∑_{p∈S_{q}} g_{p} in U(q, 10θr(q)),

and from (98), σ ∈ C^{∞}(𝔾 \ F) and σ ≥ 1 on 𝔾 \ F. Moreover, because of (100), there is a constant c_{5} such that

|X_{j}σ(x)| ≤ \frac{c_{5}}{r(x)} for all x ∈ 𝔾 \ F and j = 1, …, m_{1}.

Finally a partition of unity subordinated to the family {U_{c}(p, r(p))} is

v_{p}(x) := \frac{g_{p}(x)}{σ(x)}.

Clearly, for all x ∈ 𝔾 \ F,

∑_{p∈S} v_{p}(x) = 1,

∑_{p∈S} X_{j}v_{p}(x) = 0, for j = 1, … m_{1},

|X_{j}v_{p}(x)| ≤ \frac{c_{6}}{r(x)}, for j = 1, … m_{1} and p ∈ S. (101)

The extended function f̃ is defined as

f̃(x) := \{ \begin{matrix}f(x) & if & x ∈ F \\ ∑_{p∈S} v_{p}(x)[f(p̂) + ⟨k(p̂), πp̂(p̂^{−1} · x)⟩p̂] & if & x ∈ 𝔾 \ F \end{matrix}

where, for all q ∈ 𝔾, q̂ is one of the points in F such that dist_{c}(q, F) = d_{c}(q, q̂).

To conclude the proof we have to show that f̃ ∈ C_{𝔾}^{1}(𝔾) and that ∇_{𝔾}f̃ = k in F. Claim 1.

∇_{𝔾}f̃(a) = k(a), for all a ∈ F.

Proof of Claim 1. Fix a ∈ F, we have to prove that for d_{c}(x, a) → 0

|f̃(x) − f̃(a) − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}| = o(d_{c}(x, a)). (102)

36 BRUNO FRANCHI, RAUL SERAPIONI, AND FRANCESCO SERRA CASSANO

Now, if x ∈ F and d_{c}(x, a) → 0, simply from (i) |f̃(x) − f̃(a) − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}| = |f(x) − f(a) − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}| = o(d_{c}(x, a)). (103) If x ∈ 𝔾 \ F

|f̃(x) − f̃(a) − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}|

= |∑_{p∈S} v_{p}(x)[f(p̂) + ⟨k(p̂), πp̂(p̂^{−1} · x)⟩p̂] − f(a) − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}|

≤ |∑_{p∈S} v_{p}(x) [f(p̂) − f(a) − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}]|

+ |∑_{p∈S} v_{p}(x) [⟨k(p̂), πp̂(p̂^{−1} · x)⟩p̂ + ⟨k(a), π_{a}(a^{−1} · p̂)⟩_{a} − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}]|

= |I| + |II|.

Notice that p ∈ S_{x} ⟹ d_{c}(a, p̂) ≤ 2(1 + 2θ)d_{c}(x, a), (104) indeed

d_{c}(x, p̂) ≤ d_{c}(a, p) + d_{c}(p, p̂)
≤ 2d_{c}(a, p) ≤ 2(d_{c}(a, x) + d_{c}(x, p))
≤ 2(d_{c}(a, x) + 10θ(r(x) + r(p)))
≤ 2(d_{c}(a, x) + 40θr(x)) ≤ 2(1 + 2θ)d_{c}(x, a).

Hence, because of (104), |I| ≤ o(d_{c}(p̂, a)) = o(d_{c}(x, a)). (105) In order to estimate |II|, observe that

⟨k(a), π_{a}(a^{−1} · p̂)⟩_{a} − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a} = ⟨k(a), π_{a}(x^{−1} · p̂)⟩_{a},

hence

|⟨k(p̂), πp̂(p̂^{−1} · x)⟩p̂ + ⟨k(a), π_{a}(a^{−1} · p̂)⟩_{a} − ⟨k(a), π_{a}(a^{−1} · x)⟩_{a}|

= |⟨k(p̂), πp̂(p̂^{−1} · x)⟩p̂ + ⟨k(a), π_{a}(x^{−1} · p̂)⟩_{a}|

= |∑_{j=1}^{m_{1}}(k_{j}(p̂) − k_{j}(a))(x_{j} − p̂_{j})|

≤ d_{c}(x, p̂) ∑_{j=1}^{m_{1}} |k_{j}(p̂) − k_{j}(a)|

= o(d_{c}(x, a)),

remembering (104) and the continuity of k. Hence

|II| ≤ o(d_{c}(x, a)) ∑_{p∈S} v_{p}(x) = o(d_{c}(x, a)); (106)

(105) and (106) together with (103) give (102) and the proof of Claim 1 is completed.

Claim 2.

f̃ ∈ C_{𝔾}^{1}(𝔾).

Proof of Claim 2. Observe that f̃ is continuous in 𝔾, indeed it is Euclidean C^{∞} in 𝔾 \ F and it is 𝔾-differentiable, hence continuous, in F.

Now we fix j ∈ 1, …, m_{1}, say j = 1, and we prove the continuity of X_{j}f̃. The continuity of X_{1}f̃ in 𝔾 \ F follows from f̃ ∈ C^{∞}(𝔾 \ F); so we have only to show the continuity of X_{1}f̃ in F. Fix a ∈ F, then if x ∈ F, as proved in Claim 1,

|X_{1}f̃(a) − X_{1}f̃(x)| = |k_{1}(a) − k_{1}(x)| → 0, as d_{c}(x, a) → 0.

Then, if a ∈ F and x ∈ 𝔾 \ F,

(107)  |X_{1}f̃(x) − X_{1}f̃(a)| = |X_{1}f̃(x) − k_{1}(a)|

≤ |X_{1}f̃(x) − k_{1}(x̂)| + |k_{1}(x̂) − k_{1}(a)| . (108)

Because of d_{c}(x̂, a) ≤ d_{c}(x̂, x) + d_{c}(x, a) ≤ 2d_{c}(x, a) and the continuity of k, |k_{1}(x̂) − k_{1}(a)| vanishes as d_{c}(x, a) → 0. The proof is achieved if we prove also that |X_{1}f̃(x) − k_{1}(x̂)| → 0 as d_{c}(x, a) → 0.

|X_{1}f̃(x) − k_{1}(x̂)| = | ∑_{p∈S_{x}} X_{1}v_{p}(x) [f(p̂) + ⟨k(p̂), πp̂(p̂^{−1} · x)⟩p̂] + ∑_{p∈S_{x}} v_{p}(x) [k_{1}(p̂) − k_{1}(x̂)]|

= | ∑_{p∈S_{x}} X_{1}v_{p}(x) [f(p̂) − f(x̂) − ⟨k(x̂), πx̂(x̂^{−1} · p̂)⟩x̂]|

+ | ∑_{p∈S_{x}} X_{1}v_{p}(x)[∑_{j=1}^{m_{1}}(k_{j}(p̂) − k_{j}(x̂))(x_{j} − p̂_{j})]|

+ | ∑_{p∈S_{x}} v_{p}(x) [k_{1}(p̂) − k_{1}(x̂)]|

= |I| + |II| + |III|.

Now observe that

|I| ≤ ∑_{p∈S_{x}} |X_{1}v_{p}(x)| |f(p̂) − f(x̂) − ⟨k(x̂), πx̂(x̂^{−1} · p̂)⟩x̂|

= o(d_{c}(p̂, x̂)) ∑_{p∈S_{x}} |X_{1}v_{p}(x)|

≤ o(d_{c}(p̂, x̂))(120θ + 9)^{Q}c_{6} \frac{1}{r(x)}

= c_{7} \frac{o(d_{c}(p̂, x̂))}{d_{c}(x̂, x)}.

Recalling that, from (104) with a = x̂, we have d_{c}(x̂, p̂) ≤ 2(1 + 2θ)d_{c}(x̂, x),

|I| ≤ c_{8} \frac{o(d_{c}(p̂, x̂))}{d_{c}(p̂, x̂)} → 0, as d_{c}(x, a) → 0.

On the other hand

|II| ≤ ∑_{p∈S_{x}} (|X_{1}v_{p}(x)| ∑_{j=1}^{m_{1}} |k_{j}(p̂) − k_{j}(x̂)||x_{j} − p_{j}|)

≤ d_{c}(p, x) ∑_{p∈S_{x}} ∑_{j=1}^{m_{1}} |X_{1}v_{p}(x)| |k_{j}(p̂) − k_{j}(x̂)|

≤ d_{c}(p, x)c_{8}ϵ(d_{c}(x, a)) ∑_{p∈S_{x}} |X_{1}v_{p}(x)|

where ϵ(t) → 0 as t → 0; here we have used the continuity of k and estimate (109); eventually, remembering (101)

|II| ≤ c_{9}ϵ(d_{c}(x, a))\frac{d_{c}(p, x)}{r(x)}

≤ c_{10}ϵ(d_{c}(x, a)).

Finally, notice that from (104) it follows

d_{c}(p̂, x̂) ≤ 4(1 + θ)d_{c}(x, a). (109)

Indeed d_{c}(p̂, x̂) ≤ d_{c}(p̂, a) + d_{c}(x̂, a) ≤ 2(1 + 2θ)d_{c}(x, a) + 2d_{c}(x, a) = 4(1 + θ)d_{c}(x, a). Hence

|III| → 0 as d_{c}(x, a) → 0.

This concludes the proof of Claim 2 and of the Theorem.

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Bruno Franchi: Dipartimento di Matematica, Università di Bologna, Piazza di porta San Donato,5, 40126 Bologna, Italy

   E-mail address: franchib@dm.unibo.it

Raul Serapioni: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050, Povo (Trento) Italy

   E-mail address: serapion@science.unitn.it

Francesco Serra Cassano: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050, Povo (Trento) Italy

   E-mail address: cassano@science.unitn.it