Sub-Riemannian Currents and Slicing of Currents in the Heisenberg group $\mathbb{H}^n$

This paper aims to define and study currents and slices of currents in the Heisenberg group $\mathbb{H}^n$. Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties for them. While some such properties are similarly true in Riemannian settings, others carry deep consequences because they do not include the slices of the middle dimension $n$, which opens new challenges and scenarios for the possibility of developing a compactness theorem. Furthermore, this suggests that the study of currents on the first Heisenberg group $\mathbb{H}^1$ diverges from the other cases, because that is the only situation in which the dimension of the slice of a hypersurface, $2n-1$, coincides with the middle dimension $n$, which triggers a change in the associated differential operator in the Rumin complex.

In Section 2 we define the notion of current in the Heisenberg group and show how one can think them, only to fix the idea, as special Riemannian currents. Then we describe how a current T can be written as integral with the notion of representability by integration, denoted T " Ý Ñ T^µ T , we define its mass MpT q and show that finite mass implies representability while the two notions are equivalent if the current has compact support. Since the theory of currents has been first developed in the Riemannian setting, understandably we refer to it as much as necessary to present concepts in a linear way. Specifically, we point out when some results can be compared to the Riemannian equivalent, citing the books of Federer (see Section 4.1 in [3]), Simon ([9]) and Morgan (see Chapter 4 in [6]). Another important reference is the 2007 work by Franchi, Serapioni and Serra Cassano ( [4]). Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces. Particularly, in case we assume their support to be compact, we can work with currents of finite mass (see scheme below and figure 1); otherwise we need to consider currents with only locally finite mass (see figure 2). In Section 3, we define the notion of slices of Heisenberg currents and show some important properties for them. Slices are defined as follows: Definition (3.2). Consider an open set U Ď H n , f P LippU, Rq, t P R and T P D H,k pU q. We define slices of T the following two currents: xT, f ,t`y :" pBT q t f ą tu´B pT t f ą tuq , xT, f ,t´y :" B pT t f ă tuq´pBT q t f ă tu.
In propositions 3.5 and 3.6, we show seven properties for slices of Heisenberg currents. Specifically, Proposition 3.5 holds properties similarly true in Riemannian settings (compare with 4.2.1 in [3]) and we do not see an explicit use of the sub-Riemannian geometry in the proofs: Proposition (3.5). Consider an open set U Ď H n , T P N H,k pU q, f P LippU, Rq, and t P R. Then we have the following properties: (0) pµ T`µBT q pt f " tuq " 0 for all t but at most countably many. (1) xT, f ,t`y " xT, f ,t´y for all t but at most countably many. (2) sptxT, f ,t`y Ď f´1ttu X spt T. (3) BxT, f ,t`y "´xBT, f ,t`y.
On the other hand, the proof of Proposition 3.6, containing the remaining properties, is way more complex than in the Riemannian case and requires to explicitly work with the Rumin cohomology (see Lemma 3.11 in particular).
Proposition (3.6). Consider an open set U Ď H n , T P N H,k`1 pU q, f P LippU, Rq, t P R and k ‰ n. Then the following properties hold: (4) M pxT, f ,t`yq ď Lipp f q lim inf hÑ0`1 h µ T pU X tt ă f ă t`huq. (5) ş b a M pxT, f ,t`yqdt ď Lipp f qµ T pU X ta ă f ă buq , a, b P R. (6) xT, f ,t`y P N H,k pU q for a.e. t. Proposition 3.6 carries deep consequences for the possibility of developing a compactness theorem for currents in the Heisenberg group because it does not include the slices of the middle dimension k " n, which opens new challenges and scenarios. Furthermore, this suggests that the study of currents on the first Heisenberg group H 1 diverges from the other cases, because that is the only situation in which the dimension of the slice of a hypersurface, 2n´1, coincides with the middle dimension n, which triggers a change in the associated differential operator in the Rumin complex. Our future studies will focus, on one side, on the manipulation of the second order differential operator D in the case of the first Heisenberg group H 1 and, on the other side, on slices of currents with dimension different from n for general n ‰ 1.

PRELIMINARIES
In this section we introduce the Heisenberg group H n , its structure as a Carnot group and the standard bases of vector fields and differential forms. There exist many good references for such an introduction and we follow mainly sections 2.1 and 2.2 in [4] and sections 2.1.3 and 2.2 in [2]. We also describe briefly the Rumin cohomology and complex; more detail descriptions can be found, for example, in [8], [5] and [1].
1.1. The Heisenberg Group H n . Definition 1.1. The n-dimensional Heisenberg Group H n is defined as H n :" pR 2n`1 ,˚q, where˚is the product px, y,tq˚px 1 , with x, y, x 1 , y 1 P R n , t,t 1 P R and J "ˆ0 It is common to write x " px 1 , . . . , x n q P R n .
Furthermore, with a simple computation of the matrix product, we immediately have that One can verify that the Heisenberg group H n is a Lie group, meaning that the internal operations of product and inverse are both differentiable. In the Heisenberg group H n there are two important groups of automorphisms; the first one is the left translation τ q : H n Ñ H n , p Þ Ñ q˚p, and the second one is the (1-parameter) group of the anisotropic dilations δ r , with r ą 0: where ¨ H is the Korányi norm px, y,tq H :"`|px, yq| 4`1 6t 2˘1 4 , with px, y,tq P R 2nˆR and |¨| being the Euclidean norm.
The Korányi distance is left invariant, meaning d H pp˚q, p˚q 1 q " d H pq, q 1 q for p, q, q 1 P H n , and homogeneous of degree 1 with respect to δ r , meaning d H pδ r ppq, δ r pqqq " rd H pp, qq, for p, q P H n and r ą 0. Furthermore, the Korányi distance is equivalent to the Carnot-Carathéodory distance d cc , which is measured along curves whose tangent vector fields are horizontal.

Left Invariance and Horizontal
Structure on H n . The standard basis of vector fields in the Heisenberg group H n gives it the structure of Carnot group. By duality, we also introduce its standard basis of differential forms.
One can observe that tX 1 , . . ., X n ,Y 1 , . . . ,Y n , T u becomes tB x 1 , . . ., B x n , B y 1 , . . . , B y n , B t u at the neutral element. Another easy observation is that the only non-trivial commutators of the vector fields X j ,Y j and T are rX j ,Y j s " T , for j " 1, . . ., n. This immediately tells that all higher-order commutators are zero and that the Heisenberg group is a Carnot group of step 2. Indeed we can write its Lie algebra h as h " h 1 ' h 2 , with h 1 " spantX 1 , . . . , X n ,Y 1 , . . .,Y n u and h 2 " spantT u.
Conventionally one calls h 1 the space of horizontal and h 2 the space of vertical vector fields. The vector fields tX 1 , . . . , X n ,Y 1 , . . .,Y n u are homogeneous of order 1 with respect to the dilation δ r , r P R`, i.e., X j p f˝δ r q " rX j p f q˝δ r and Y j p f˝δ r q " rY j p f q˝δ r , where f P C 1 pU, Rq, U Ď H n open and j " 1, . . . , n. On the other hand, the vector field T is homogeneous of order 2, i.e., It is not a surprise, then, that the homogeneous dimension of H n is Q " 2n`2.
The vector fields X 1 , . . ., X n ,Y 1 , . . .,Y n , T form an orthonormal basis of h with a scalar product x¨,¨y. In the same way, X 1 , . . . , X n ,Y 1 , . . . ,Y n form an orthonormal basis of h 1 with a scalar product x¨,¨y H defined purely on h 1 . :" X j for j " 1, . . . , n, W n`j :" Y j for j " 1, . . . , n, W 2n`1 :" T.
In the same way, the point px 1 , . . ., x n , y 1 , . . . , y n ,tq will be denoted as pw 1 , . . ., w 2n`1 q. Definition 1.5. Consider the dual space of h, Ź 1 h, which inherits an inner product from h. By duality, one can find a dual orthonormal basis of covector fields tω 1 , . . . , ω 2n`1 u in Ź 1 h such that xω j |W k y " δ jk , for j, k " 1, . . . , 2n`1, where W k is an element of the basis of h. Such covector fields are differential forms in the Heisenberg group.
The orthonormal basis of Ź 1 h is given by tdx 1 , . . ., dx n , dy 1 , . . . , dy n , θ u, where θ is called contact form and is defined as px j dy j´y j dx j q. Example 1.6. As a useful example, we show here that the just-defined bases of vectors and covectors behave as one would expect when differentiating. Specifically, consider f : U Ď H n Ñ R, U open, f P C 1 pU, Rq, then one has: Definition 1.7. We define the sets of k-dimensional vector fields and differential forms, respectively, as: and Ω k " ľ k h :" spantdw i 1^¨¨¨^d w i k u 1ďi 1 ď¨¨¨ďi k ď2n`1 .
The same definitions can be given for h 1 and produce the spaces Ź k h 1 and Ź k h 1 .
Next we give the definition of Pansu differentiability for maps between Carnot groups G and G 1 . After that, we state it in the special case of G " H n and G 1 " R.
uniformly for p in compact subsets of U .
Notation 1.12 (see 2.12 in [4]). Sets of differentiable functions can be defined with respect to the P-differentiability. Consider U Ď G and V Ď G 1 open, then C 1 H pU,V q is the vector space of continuous functions f : U Ñ V such that the P-differential d H f is continuous.
To conclude this part, we define the Hodge operator which, given a vector field, returns a second one of dual dimension and orthogonal to the first.

Rumin
Cohomology in H n . The Rumin cohomology is the equivalent of the Riemann cohomology but for the Heisenberg group. Its complex is given not by one but by three operators, depending on the dimension. Definition 1.14. Consider 0 ď k ď 2n`1 and recall Ω k from Definition 1.7. We denote: Notation 1.15 (see 2.1.8 and 2.1.10 in [1]). We denote L the operator Furthermore we remind that, if γ P Ω k´1 , we can consider the equivalence class where we write tγ^θ u " tγ^θ ; γ P Ω k´1 u for short. The equivalence is given by β Þ Ñ pβ q |Ź k h 1 .
In particular, L is an isomorphism (see 2 in [8]) and we can denote Notation 1.16. We denote by rαs I k an element of the quotient Ω k I k and ω | J k an element of J k whenever ω P D k pU q. We will use this second definition later on. Definition 1.17 (Rumin complex). The Rumin complex, due to Rumin in [8], is given by where d is the standard differential operator and, for k ă n, while, for k ě n`1, d Q :" d | J k . The second order differential operator D is defined as These three different differential operators are at times denoted with the same syntax d c or d pkq c , when they act on k-forms (see Theorem 11.40 in [5] or Proposition B.7 in [1]).

CURRENTS IN THE HEISENBERG GROUP
In this section we first define the notion of current in the Heisenberg group and expose its relationship with Riemannian currents. Then we describe how currents can be written as integrals with the notion of representability by integration, define the mass of a current in H n and show that finite mass implies representability and the two notions are equivalent if the current has compact support. Last, we classify currents into subspaces depending on the integration properties of themselves and their boundaries and we work with currents with finite mass if the support is compact (see figure 1), while we consider currents with only locally finite mass otherwise (see figure 2). In Riemannian geometry there are different kind of currents and the correlation between the different definitions is well known since Federer (see Section 4.1 in [3]); useful references are also the works of Simon ([9]) and Morgan (see Chapter 4 in [6]). Finally, for the Heisenberg group specifically, an important reference is the 2007 work by Franchi, Serapioni and Serra Cassano ( [4]).
Definition 2.1 (see 5.8 in [4]). Consider an open set U Ď H n . We call D k H pU q the space of compactly supported smooth sections on U of, respectively, Ω k I k , if 1 ď k ď n, and J k , if n`1 ď k ď 2n`1. These spaces are topologically locally convex. For convenience, we call the elements of D k H pU q Rumin or Heisenberg differential forms. Furthermore, we call Rumin or Heisenberg current any continuous linear functional from the space D k H pU q to R and we denote their set as D H,k pU q. We just saw in Definition 2.1 that the Rumin currents are defined, for low dimensions, on quotient spaces. Nevertheless it is possible, to fix the ideas, to think about Rumin differential forms as a subset of the standard differential forms and so write D k H pU q Ď D k pU q for simplicity. In the same way, we can think about Rumin currents as a subset of the Euclidean currents. Indeed, any Rumin current T P D H,k pU q can be identified with an Euclidean k-current r T P D k pU q by setting, for ω P D k pU q: . Consider an open set U Ď H n and T P D k H pU q. The support of a current T is defined as 2.1. Representability by Integration and Masses in H n . In the study of currents, it is often useful to be able to write a current as an integral. The first notion we see that allows us to do so is representability by integration. After that we define the mass of currents in H n and show that finite mass implies representability and the two notions are equivalent if the current has compact support. Since the theory of currents has been first developed in the Riemannian setting, understandably we refer to it as much as necessary to present concepts in a linear way. Specifically, we point out when some results can be compared to the Riemannian equivalent, citing the books of Federer ( where v is integrable if and only if the distribution associated to it is so. By duality, for 0 ď k ď 2n`1, .
Note that, by Theorem 2.9 in [4], the spaces H Ź k 's are the spaces of the Rumin cohomology. So the spaces of vector fields H Ź k 's are the dual of the Rumin differential forms. Definition 2. 4. Consider an open set U Ď H n and T P D H,k pU q. We say that T is representable by integration, and we write T " Ý Ñ T^µ T , if there exist µ T a Radon measure over U and a vector Before we define the mass of a current, a clarification is necessary. In the standard theory of currents there are two different notion of mass for a current: one made using the comass of differential forms (see 4.3 in [6] and 4.1.7 in [3]) and one using the norm given by the inner product of differential forms (see, for instance, 2.6Ch6 in [9]). This is still true in our case.
Definition 2.5 (mass of a current by the comass in H n ). Consider an open set U Ď H n and T P D H,k pU q. Denote the mass of a current T defined by the comass as: . Other notations for the comass in the literature are Mpωq and ωppq . Definition 2.6 (mass of a current by the scalar product in H n , see 5.12 in [4]). Consider an open set U Ď H n and T P D H,k pU q. Denote the mass of a current T defined by the scalar product as: with |ω| " a xω, ωy, where x¨,¨y is the Riemannian scalar product that makes the differential forms dx j , dy j 's and θ orthonormal.
The comass is smaller or equal than the scalar product norm (see also 2.6Ch6 in [9]), which means that the mass defined with the comass is bigger or equal than the one defined with the scalar product: mpT q ď MpT q for all T P D H,k pU q. Finally we state the correlation between mass and currents representable by integration (compare with 4.1.7 in [3] and 2.8Ch6 in [9]). The proof is based on Riesz Representation Theorem and it is not dissimilar from the same proof in the Riemannian setting. In particular (compare with 2.6Ch6 and 4.14Ch1 in [9]), if MpT q ă 8, then both masses are finite, µ T is unique, Ý Ñ T " Ý Ñ T m a.e. and µ T pU q " MpT q " mpT q " µ T,m pU q.
because T has compact support.

2.2.
Classification of Sub-Riemannian Currents in H n . Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces. Particularly, in case we assume their support to be compact, we can work with currents of finite mass (see figure 1); otherwise we need to consider currents with only locally finite mass (see figure 2).
Definition 2.10 (see 5.19 in [4]). Consider an open set U Ď H n , a current T P D H,k pU q and 1 ď k ď 2n`1. We call Heisenberg boundary of T the pk´1q-dimensional Heisenberg current denoted BT (or sometimes B H T ) and defined as: where ω P D k´1 H pU q. Definition 2.11. Consider an open set U Ď H n and 1 ď k ď 2n`1. We define the space of currents with compact support as E H,k pU q :" T P D H,k pU q { spt T compact ( . Furthermore, we can define the spaces of currents with finite mass as By Corollary 2.9, we can immediately characterise the spaces as follows:  [4]). Consider 1 ď k ď n. A subset S Ď H n is a H-regular kcodimensional surface if for all p P S there exists a neighbourhood U of p and a function f : U Ñ R k , f P C 1 H pU, R k q, such that ∇ H f 1^¨¨¨^∇H f k ‰ 0 on U and S XU " t f " 0u. Definition 2.14 (see 5.1 in [4]). Consider S Ď H n and S k 8 the spherical Haussdorff measure defined in Subsection 2.1 in [4]. We say that S is a k-dimensional H-rectifiable set if T P E H,k pU q { T pωq " where U T is an H-rectifiable k-dimensional set oriented (up to a set of measure zero) by Ý Ñ T , a µ k -a.e. unit k-vector in H Ź k , ρ is a positive integer multiplicity s.t. ş U T Xspt T ρppqdµ k ă 8 and µ k :" Then we define the space of space of integral H-rectifiable currents as I H-rect,k pU q :" T P R H-rect,k pU q { BT P R H-rect,k´1 pU q ( Ď R H-rect,k pU q. This also immediately implies that I H-rect,k pU q Ď N H,k pU q.
Proof. The proof is a simple computation. Consider T P R H-rect,k pU q, then: |MpT q| " sup Proof. The first equality in the statement comes from Proposition 2.7. For the second equality, by Proposition 2.16, we know that T P R H-rect,k pU q implies T " Ý Ñ T^µ T . At the same time, T P R H-rect,k pU q says that we can write By uniqueness of the representation by integration, that comes from Riesz Representation Theorem, we have that µ T " ρ µ k pU T X spt T q , i.e., µ T pU q " We remind that a C 1 -Euclidean regular k-surface can be written as S " CpSq Y pSzCpSqq where, for n`1 ď k ď 2n`1, S k`1 8 pCpSqq " 0 and SzCpSq is a H-regular surface (see page 195 in [4]). For this reason, when n`1 ď k ď 2n`1,

SLICING OF CURRENTS IN THE HEISENBERG GROUP
In this section we define the notion of slices of Heisenberg currents and show, in propositions 3.5 and 3.6, seven important properties. Proposition 3.6, in particular, carries deep consequences for the possibility of developing a compactness theorem for currents in the Heisenberg group because it does not include the slices of the middle dimension k " n. Furthermore, this suggests that the study of currents on the first Heisenberg group H 1 diverges from the other cases, because that is the only situation in which the dimension of the slice of a hypersurface, 2n´1, coincides with the middle dimension n, which triggers a change in the associated differential operator in the Rumin complex. The most important references for the Riemannian case are sections 4.1.7 and 4.2.1 in [3] and the matching sections in [6]. ‚ If f P D 0 H pU q " C 8 pU q, T P D H,k pU q and ω P D k H pU q, then pT f qpωq :" T p f ωq.
H pU q, m ď k, T P D H,k pU q and ω P D k´m H pU q, then pT ϕqpωq :" T pϕ^ωq.
‚ If A Ď H n Borel set, χ A : H n Ñ t0, 1u and T P R H,k pU q, then ‚ If T P D H,k pU q is representable by integration, T " Ý Ñ T^µ T , and a function f : U Ñ R is such that ş | f |dµ T ă 8, then

Definition 3.2.
Consider an open set U Ď H n , f P LippU, Rq, t P R and T P D H,k pU q. We define slices of T the following two currents: xT, f ,t`y :" pBT q t f ą tu´B pT t f ą tuq , xT, f ,t´y :" B pT t f ă tuq´pBT q t f ă tu.
It is important to notice that, considering an open set U Ď H n , a function f P C 8 pU q and a current T P R H,k pU q`resp. R H-rect,k pU q or R H,k pU q˘, we cannot imply that T f P R H,k pU q`resp. R H-rect,k pU q or R H,k pU q˘. The reason is that, applying a smooth function to the current, without further hypotheses, we cannot always expect the current mass to remain finite. Nevertheless, something can still be said.
Note that the following lemma contains three statement each (one in R H,k pU q, one in R H-rect,k pU q and one in R H,k pU q); they are written together as the proofs are basically the same.

Lemma 3.3.
Consider an open set U Ď H n , A Ď H n a Borel set and T P R H,k pU q`resp. R H-rect,k pU q or R H,k pU q˘. Then T χ A P R H,k pU q`resp. R H-rect,k pU q or R H,k pU q˘.
The proof of this lemma is a one-line application of the definitions. Then Proof. We can compute directly, using the linearity of the definition of currents, xT, f ,t`y " pBT q t f ą tu´B pT t f ą tuq " pBT q pH n zt f ď tuq´B pT pH n zt f ď tuqq " BT´pBT q t f ď tu´B pT´T t f ď tuq " B pT t f ď tuq´pBT q t f ď tu.
The same can be done for xT, f ,t´y.

Properties of Slices.
In the next two propositions, we show seven properties for slices of Heisenberg currents. Specifically, Proposition 3.5 holds properties similarly true in Riemannian settings (compare with 4.2.1 in [3]) and indeed we do not see an explicit use of the sub-Riemannian geometry in the proofs. On the other hand, Proposition 3.6, containing the remaining properties, requires k ‰ n, which carries deep consequences, especially when n " 1. Furthermore, the proof of Proposition 3.6 is way more complex than in the similar Riemannian case and requires to explicitly work with the Rumin cohomology. This work follows the Riemannian theory of Federer, in particular section 4.2.1 in [3].

Proposition 3.5.
Consider an open set U Ď H n , T P N H,k pU q, f P LippU, Rq, and t P R. Then we have the following properties: (0) pµ T`µBT q pt f " tuq " 0 for all t but at most countably many. (1) xT, f ,t`y " xT, f ,t´y for all t but at most countably many. (2) sptxT, f ,t`y Ď f´1ttu X spt T.
Proof. Property (0) holds as a general statement for measures. By Lemma 3.4, xT, f ,t`y " B pT t f ď tuq´pBT q t f ď tu.
Consider now T t f " tu and notice that T t f " tu P R H,k pU q by Lemma 3.3, meaning that T t f " tu is a current representable by integration. In particular, by property p0q, pT t f " tuq p‹q " ż t f "tu x‹| Ý Ñ T ydµ T " 0, for all t but at most countably many.
In the same way, pBT q t f " tu P R H,k´1 pU q by hypothesis and so, again by property (0), for all t but at most countably many. So we can write that, for all t but at most countably many, This proves property p1q.
Next we prove property p3q, leaving property p2q as last. We have On the other hand xBT, f ,t`y "´prB pBT qs t f ą tu´B ppBT q t f ą tuqq So also property p3q is verified. Only property p2q is left, namely that sptxT, f ,t`y Ď f´1ttu X spt T . Recalling Definition 2.2, p P sptxT, f ,t`y if and only if there exists a neighbourhood U p of p and a differential form ω P D k´1 H pU q such that xT, f ,t`ypωq ‰ 0 and spt ω Ď U p . This is the same as asking (3.1) rpBT q t f ą tu´B pT t f ą tuqs pωq ‰ 0.
By contradiction, suppose that p R spt T , which means that there exists another neighbourhood of p,Ũ p , such thatŨ p X spt T " ∅ ( figure 3). Shrinking U p andŨ p if needed (which means we may also restrict ω accordingly), we can assume U p "Ũ p , and so spt ω X spt T " ∅. Note then that, for α P D k´1 H pU q, BT pαq " T pd c αq (where d c is the Rumin complex operator in general dimension, see Definition 1.17), hence spt BT Ď spt T . Then spt ω X spt BT " ∅.
But this is a contradiction with equation (3.1), so we have that p P spt T . Consider now p P sptxT, f ,t`y as above and, by contradiction again, suppose than p R f´1ttu: By hypothesis there exists a neighbourhood U p of p and a differential form ω P D k´1 H pU q such that spt ω Ď U p and equation (3.1) holds. In particular, we can choose U p so that rpBT q t f ą tus pωq " pBT q`χ t f ątu ω˘" pBT q pωq " T pd c ωq .
In a similar way, So xT, f ,t`ypωq " rpBT q t f ą tu´B pT t f ą tuqs pωq " 0 which is a contradiction. If f ppq ă t, then spt ω Ď U p Ď t f ă tu Ď t f ď tu and we have rB pT t f ď tuqs pωq " pT t f ď tuq pd c ωq " T pχ t f ďtu d c ωq " T pd c ωq and rpBT q t f ď tus pωq " pBT q`χ t f ďtu ω˘" pBT q pωq " T pd c ωq .
Again, using Lemma 3.4, xT, f ,t`ypωq " rB pT t f ď tuq´pBT q t f ď tus pωq " 0 which is a contradiction. This complete the proof.
As the proof showed, the geometry of the Heisenberg group and the Rumin complex, although present, did not play a role in the previous properties. Now we show further properties for which the Rumin cohomology does play a bigger role.
Proposition 3. 6. Consider an open set U Ď H n , T P N H,k`1 pU q, f P LippU, Rq, t P R and k ‰ n. Then the following properties hold: The case k " n present several differences from what we show here and, although work in that direction is ongoing, one can very easily expect differences in the final result. This comes with deep consequences as these properties are meant to be tools to help develop a compactness theorem for currents in the Heisenberg group. In detail, this corroborates that the Riemannian approach is not effective here and that new ideas are necessary. Furthermore, this also suggests that the study in the first Heisenberg group H 1 diverges from the other cases' because, when n " 1, then k " np" 1q is the most important situation.
The first point is the most complicated to prove. For this reason we first contruct some machinary and show some lemmas. |s´t|´|s´pt`hq|`h 2h . One can observe that Proof. The computation of γ h˝f follows immediately from the definition. Then, for p, q P U and considering t ă f ă t`h, This implies that γ h˝f P LippU, Rq and, since its Lipschitz constant is the smallest for which the inequality holds, also Lippγ h˝f q ď Lipp f q h is verified. Proof. Let's start by considering M pxT, f ,t`y´pBT q pγ h˝f q`B pT pγ h˝f qqq "M`pBT q χ t f ątu´B`T χ t f ątu˘´p BT q pγ h˝f q`B pT pγ h˝f qq" Let's estimate the two terms independently. By construction χ t f ątu´γh˝f " 0 on t f ě t`hu and χ t f ątu´γh˝f ď 1 on tt ă f ă t`hu, so Then, for ω P D k´1 H pU q,ˇ`p by monotone convergence theorem, which allows the limit over the integral. For the second term we have:ˇ`B`T`γ Then we have thatˇˇM`p BT q`χ t f ątu´γh˝f˘˘"ˇs Putting the two terms together, we get |M pxT, f ,t`y´pBT q pγ h˝f q`B pT pγ h˝f qqq| This also means that M pxT, f ,t`y´pBT q pγ h˝f q`B pT pγ h˝f qqq Ý ÝÝ Ñ hÑ0 0.

Finally we observe
MpxT, f ,t`yq ďM pxT, f ,t`y´pBT q pγ h˝f q`B pT pγ h˝f qqq M ppBT q pγ h˝f q´B pT pγ h˝f qqq and, passing to the lim inf for h Ñ 0, we obtain the claim. Lemma 3.9. Consider an open set U Ď H n , f P LippU, Rq, t P R, h ą 0 fixed and consider the function γ h defined in Lemma 3.7. Then we can approximate γ h˝f uniformly by functions g i P C 8 pU, Rq (notationally g i Ñ γ h˝f ), so that spt dg i Ď tt ă f ă t`hu and lim iÑ8 Lippg i q " Lippγ h˝f q.
Proof. By density of smooth functions, we can approximate γ h˝f uniformly by smooth function g i P C 8 pU, Rq and, since γ h˝f is smooth and locally constant out of tt ă f ă t`hu, it follows that g i is locally constant out of tt ă f ă t`hu as well and so that spt dg i Ď tt ă f ă t`hu. To prove the limit, we see that, for p, q P U , |g i ppq´g i pqq| ď |g i ppq´γ h˝f ppq|`|γ h˝f ppq´γ h˝f pqq|`|γ h˝f pqq´g i pqq| M ppBT q pγ h˝f q´B pT pγ h˝f qq´rpBT q g i´B pT g i qsq " lim iÑ8 M ppBT q pγ h˝f´gi q´B pT pγ h˝f´gi qqq "0 since g i Ñ γ h˝f . Then M ppBT q pγ h˝f q´B pT pγ h˝f qqq ďM``BT˘pγ h˝f q´B pT pγ h˝f qq´rpBT q g i´B pT g i qsM ppBT q g i´B pT g i qq .
Passing to the limit for i Ñ 8, we obtain the claim.
So far we could work without explicitely using the Rumin complex operators. Now this is no more possible, as the following lemma shows. Lemma 3.11. Consider an open set U Ď H n , T P D H,k`1 pU q, ω P D k H pU q and the functions g i P C 8 pU, Rq defined in Lemma 3.9. Also recall notations 1.15 and 1.16. Then rpBT q g i´B pT g i qs pωq " if k ă n, T´d p1q g i^p ω`Lpωq^θ q`d pn`1q ppLpg i ωq´g i Lpωqq^θ q¯, rωs I n P D n H pU q " Ω n I n , if k " n, Tˆ´d p1q g i^ω¯| J k`1˙, "d pn`1q pg i ωq´g i d pn`1q ω`d pn`1q pLpg i ωq^θ q´g i d pn`1q pLpωq^θ q "d p1q g i^ω`" d pnq pLpg i ωqq´g i d pnq pLpωqq ı^θ p´1q n´1 rLpg i ωq´g i Lpωqs^d p2q θ "d p1q g i^ω`" d pnq pLpg i ωqq´´d pnq pg i Lpωqq´d p1q g i^L pωq¯ı^θ p´1q n´1 rLpg i ωq´g i Lpωqs^d p2q θ "d p1q g i^ω`d p1q g i^L pωq^θ`d pnq pLpg i ωq´g i Lpωqq^θ p´1q n´1 pLpg i ωq´g i Lpωqq^d p2q θ "d p1q g i^p ω`Lpωq^θ q`d pn`1q ppLpg i ωq´g i Lpωqq^θ q .
This completes the proof of the lemma.

Lemma 3.12.
Consider an open set U Ď H n , T P R H,k`1 pU q, ω P D k H pU q, k ‰ n and the functions g i P C 8 pU, Rq defined in Lemma 3.9. Then rpBT q g i´B pT g i qs pωq ď Lippg i q pT spt dg i q¨2 n ÿ j"1 dw j^ω‚ .
Proof. For k ă n, by Lemma 3.11, rpBT q g i´B pT g i qs pωq " T prdg i^ω s I k`1 q " ż UXspt dg i For k ą n, by Lemma 3.11 again, we have a similar expression: rpBT q g i´B pT g i qs pωq " T´pdg i^ω q | J k`1" ż Recall Notation 1.4 and note that, as in Example 1.6, dg i " ř 2n`1 j"1 W j g i dw j . If k ą n, then ω P D k H pU q is of the form ω " dw 2n`1^ω 1 , ω 1 P Ω k´1 (see J k at Definition 1.14). We note that |∇ H g i | ď Lippg i q and so W j g i ď Lippg i q for all j " 1, . . . , 2n. Indeed, using Definitions 1.10 and 1.11, |W j g i pp 0 q| ď |∇ H g i pp 0 q| " |d H g i p 0 ppq| " lim rÑ0`| g i pp 0˚δr ppqq´g i pp 0 q| r .
In particular we can choose p so that d H pp 0 , p 0˚δr ppqq " r. Then we denote q " p 0˚δr ppq, which gives d H pp 0 , qq " r and  This proves property (4). The other two properties follow quickly. To prove property (5) we proceed as in 4.11 in [6]. Consider Fptq " µ pU X t f ă tuq, an increasing monotone function with derivative almost everywhere.
Lipp f qµ T pU X ta ă f ă buq " Lipp f q pµ T pU X t f ă buq´µ T pU X t f ď auqq This proves property (5). By Proposition 2.7 and since T P N H,k`1 pU q, we have that µ T pU X ta ă f ă buq ă 8. Then, by property (5), M pxT, f ,t`yq ă 8 for a.e. t.