Poincar{\'e} and Sobolev inequalities for differential forms in Heisenberg groups

Poincar{\'e} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.


Sobolev and Poincaré inequalities for differential forms
The Sobolev inequality in R n states that if u is a compactly supported function, then u q C p,q,n du p whenever 1 p, q < +∞, where du is the differential of u (which is a 1-form). A local version, for functions supported in the unit ball, holds under the weaker assumption The Poincaré inequality is a variant for functions u defined, but not necessarily compactly supported, in the unit ball B. It states that there exists a real number c u such that u − c u q C p,q,n du p .
Alternatively, for a given exact 1-form ω on B, there exists a function u on B such that du = ω on B and such that u q C p,q,n ω p .
This suggests the following generalization for higher-degree differential forms in Riemannian manifolds.
Let M be a Riemannian manifold with or without boundary. We say that a global Poincaré inequality holds on M if there exists a positive constant C = C(M, p, q) such that for every exact h-form ω on M, belonging to L p , there exists an (h − 1)-form φ such that dφ = ω and φ q C ω p .
Shortly, we shall say that Poincaré p,q (h) holds.
A global Sobolev inequality holds on M if for every exact compactly supported h-form ω on M, belonging to L p , there exists a compactly supported (h − 1)-form φ such that dφ = ω and φ q C ω p .
Again, we shall say that Sobolev p,q (h) holds.
In both statements, the assumption that given forms are exact is there to separate the topological problem (whether a given closed form is exact) from the analytical one (whether a primitive can be upgraded to one that satisfies estimates).

.1 and equation (169)]).
However, for more general Euclidean domains, the validity of the Poincaré inequality is sensitive to irregularities of boundaries. One way to eliminate such a dependence is to allow a loss on domain. For the case p = 1 in the Euclidean setting, we refer the reader to [4].
Say an interior Poincaré inequality Poincaré p,q (h) holds on M if for every small enough r > 0 and large enough λ 1, there exists a constant C = C(M, p, q, r, λ) such that for every x ∈ M and every exact h-form ω on B(x, λr ), belonging to L p , there exists an (h − 1)-form φ on B(x, r ) such that dφ = ω on B(x, r ) and φ L q (B(x,r )) C ω L p (B(x,λr )) . (1) By interior Sobolev inequalities, we mean that, if ω is supported in B(x, r ), then there exists φ supported in B(x, λr ) such that dφ = ω and φ L q (B(x,λr )) C ω L p (B(x,r )) .
It turns out that in several situations, the loss on domain is harmless. This is the case for L q, p -cohomological applications; see [47].
Let us comment on the terminology. Due to the loss on domain, inequality (1) provides no information on the behavior of differential forms near the boundary of their domain of definition. This is why we speak of an interior Poincaré inequality.

Contact manifolds
A contact structure on an odd-dimensional manifold M is a smooth distribution of hyperplanes H , which is maximally non-integrable in the following sense: if θ is a locally defined smooth 1-form such that H = ker(θ ), then dθ restricts to a non-degenerate 2-form on H , i.e., if 2n + 1 is the dimension of M, then θ ∧ (dθ ) n = 0 on M (see [41,Proposition 3.41]). A contact manifold (M, H ) is the data of a smooth manifold M and a contact structure H on M. Contact diffeomorphisms (also called contactomorphisms; see Definition 2.13) are contact structure preserving diffeomorphisms between contact manifolds. The prototype of a contact manifold is the Heisenberg group H n , the simply connected Lie group whose Lie algebra is the central extension with bracket h 1 ⊗ h 1 → h 2 = R being a non-degenerate skew-symmetric 2-form. The contact structure is obtained by left-translating h 1 . According to a theorem by Darboux, every contact manifold is locally contactomorphic to H n . The Heisenberg Lie algebra admits a one-parameter group of automorphisms δ t , which are counterparts of the usual Euclidean dilations in R n . Thus, differential forms on h split into two eigenspaces under δ t . Therefore, the de Rham complex lacks scale invariance under these anisotropic dilations.
A substitute for the de Rham complex, which recovers scale invariance under δ t , has been defined by Rumin [50]. It makes sense for arbitrary contact manifolds (M, H ) and it is invariant under contactomorphisms.
Let h = 0, . . . , 2n + 1. Rumin's substitute for smooth differential forms of degree h is the smooth sections of a vector bundle E h 0 . If h n, E h 0 is a sub-bundle of h H * . If h n, E h 0 is a sub-bundle of h H * ⊗ (T M/H ). Rumin's substitute for de Rham's exterior differential is a linear differential operator d c from sections of E h 0 to sections of E h+1 0 such that d 2 c = 0. We stress that the operator d c has order 2 when h = n and order 1 otherwise.
This phenomenon will be a major issue in the proofs of our results and will affect the choice of the exponents p, q in our inequalities.
The data of (M, H ) equipped with a scalar product g, defined on sub-bundle H only, is called a sub-Riemannian contact manifold and we shall write (M, H, g). The scalar product on H determines a choice of a local contact form θ and hence a norm on the line bundle T M/H . Therefore E h 0 are endowed with a scalar product. Using θ ∧ (dθ ) n as a volume form, one gets L p -norms on spaces of smooth Rumin differential forms.
In any sub-Riemannian contact manifold (M, H, g), we can define a sub-Riemannian distance d M (see e.g., [43]) inducing on M the same topology of M as a manifold. In particular, Heisenberg groups H n can be viewed as sub-Riemannian contact manifolds. If we choose on the contact sub-bundle of H n a left-invariant metric, it turns out that the associated sub-Riemannian metric is also left-invariant. It is customary to call this distance in H n a Carnot-Carathéodory distance.
Poincaré and Sobolev inequalities for differential forms make sense on contact sub-Riemannian manifolds: merely replace the exterior differential d with d c . All left-invariant sub-Riemannian metrics on the Heisenberg group are bi-Lipschitz equivalent, and hence we may refer to the sub-Riemannian Heisenberg group without referring to a specific left-invariant metric: if a Poincaré inequality holds for some left-invariant metric, it holds for all of them. On the other hand, in the absence of symmetry assumptions, large scale behaviors of sub-Riemannian contact manifolds are diverse. Examples illustrating this phenomenon will be given in § 7.

Results on Poincaré and Sobolev inequalities
In this paper, we prove global H-Poincaré and H-Sobolev inequalities and interior H-Poincaré and H-Sobolev inequalities in Heisenberg groups, where the prefix H is meant to stress that the exterior differential is replaced with Rumin's exterior differential d c . The range of parameters differs slightly from the Euclidean case due to the fact that d c has order 2 in middle dimension. Let h ∈ {0, . . . , 2n + 1}. We say that assumption E(h, p, q, n) holds if 1 < p q < ∞ satisfy Say that assumption I (h, p, q, n) holds if 1 < p q < ∞ satisfy if h = n + 1. Precise formulations of interior Poincaré and Sobolev inequalities are given in § 5. Remark 1.3. We stress that the core of the present paper is the proof of the interior inequalities of Theorem 1.2. In fact, since p > 1, the global estimates of Theorem 1.1 are more or less straightforward consequences of the L p − L q continuity of singular integrals of potential type (see § 1.6).
Here is a simple consequence of these results. Combining both theorems with results from [47], we get the following corollary.
Under assumption E(h, p, q, n), the q, p -cohomology in degree h of H n vanishes.
Our third result is the construction of a smoothing homotopy on general contact manifolds. Under a bounded geometry assumption, uniform estimates can be given (precise definitions of bounded geometry contact manifolds, as well as of associated Sobolev spaces W j, p , will be given in § 4.2). Theorem 1.5. Let k 3 be an integer. Let (M, H, g) be a 2n + 1-dimensional sub-Riemannian contact manifold of bounded C k -geometry. Under assumption I (h, p, q, n), there exist operators S M and T M on h-forms on M, which are bounded from W j, p to W j,q for all 0 j k − 1, and such that Furthermore, S M and T M are bounded from W j−1, p to W j, p if j 1 (resp. from W j−2, p to W j, p if j 2 and degree h = n + 1).
We stress that the 'approximate homotopy formula' (4) has no consequences for the cohomology of M. The iteration of the process yields an operator S M , which is bounded from L p to W k−1,q , and still acts trivially on cohomology. For instance, it is possible to replace a closed form with a much more regular differential form (up to adding a controlled exact form).

State of the art
This paper is part of a larger project aimed to prove ( p, q)-Poincaré and Sobolev inequalities in Heisenberg groups when 1 p < q ∞. Thus it seems convenient to point out the different cases we have to deal with. Let us restrict ourselves for a while to Euclidean spaces R n and Heisenberg groups H n . The first fundamental distinction is the following: (i) global inequalities (i.e., inequalities on all the space R n or H n ); (ii) interior inequalities (for instance on Carnot-Carathéodory balls).
For each one of the above geometric assumptions, we must distinguish between (iii) the case p = 1 and (iv) the case p > 1.
In the scalar case, ( p, q)-Poincaré and Sobolev inequalities are well understood both in Euclidean spaces and in Heisenberg groups for all p 1. Consider now differential forms of higher degree.
For the case p = 1, global inequalities in R n (Gagliardo-Nirenberg inequalities for differential forms) have been proved by Bourgain & Brezis [15] and Lanzani & Stein [36] via a suitable identity for closed differential forms and relying on careful estimates for divergence-free vector fields. Thanks to the counterpart of this identity proved by Chanillo & van Schaftingen in homogeneous groups [18], similar global inequalities for differential forms in H n were proved in [3]. We stress that in [3], algebra plays an important role precisely in the proof of the identities for closed forms. Therefore apart from Heisenberg groups, only a handful of more general nilpotent groups have been treated [11].
Interior inequalities when p = 1 use the estimate of [3] combined with an approximate homotopy formula introduced in the present paper, but require a new different argument to control the commutator between Rumin's exterior differential (or de Rham's exterior differential in R n ) and multiplication by a cut-off function. These inequalities are proved for Heisenberg groups in [6] and in [4] for Euclidean spaces. Note that in the Heisenberg group case, one more algebraic obstacle shows up, averages of L 1 forms; see [49].
Consider now the case p > 1. In the Euclidean setting, interior Poincaré inequalities for p > 1 are proved in [33]. However, the arguments of [33] do not extend to Heisenberg groups. Thus, the core of the present paper is the proof of interior Poincaré and Sobolev inequalities in H n when p > 1. Indeed, as we shall point out later (see Remark 1.3), when p > 1, global inequalities in H n (as well as in R n ) are more or less straightforward.
On the contrary, interior inequalities require a different more sophisticated argument (see § 1.7 for a gist of our proof). At the same time, the techniques introduced in the present paper differ substantially from those of [3] for global inequalities for p = 1.
The case when q = ∞ can be obtained by duality, and this will appear in [5]. We refer the reader also to [7] for endpoint inequalities in Orlicz spaces.
For more general sub-Riemannian spaces, the strategy is to reduce to large scale invariants (see § 7). For this, one must pass via interior inequalities and a global smoothing procedure, like in Theorems 1.2 and 1.5. In particular, in the present paper and in [6] we deal with a special class of sub-Riemannian manifolds, the sub-Riemannian contact manifold of bounded C k -geometry as in Definition 4.9.

Open questions
Keeping in mind the analogous inequalities in the scalar case, the following (still open) questions naturally arise.
(1) Do Poincaré and Sobolev inequalities hold without loss of the domain for some family of specific domains as, e.g., for metric balls associated with a left-invariant homogeneous distance?
(2) Since Heisenberg groups provide the simplest non-commutative instance of arbitrary Carnot groups (connected, simply connected stratified nilpotent groups; see [45]), the following question naturally arises: How much of these results do extend to more general Carnot groups?
Let us make a few comments about the previous questions.
(1) When dealing with scalar functions it is possible to obtain H-Poincaré p,q inequalities on Carnot-Carathéodory balls without loss on the domain and the argument relies on the so-called Boman chain condition (see, e.g., [22,24]). However, it is not clear at all how to extend this technique to differential forms.
(2) The argument used in this paper relies on an appropriate approximate homotopy formula (see point (2) in § 1.7). It is reasonable to expect that the construction of the approximate homotopy operator could be generalized to more general Carnot groups using the construction carried out in [8,48] to prove a compensated compactness result (see formulas (37) and (38) in [8]). However, for Carnot groups, we expect only unsharp estimates due to the crucial role of a fundamental solution of a 0-order Laplace operator mixing up components of forms of different homogeneity.
Further comments related to this question can be found in Remark 5.22, where specific examples in more general Carnot groups are given.
Let us give now a sketch of the proofs.

Global homotopy operators
The most efficient way to prove a Poincaré inequality is to find a homotopy between identity and 0 on the complex of differential forms, i.e., a linear operator K that raises the degree by 1 and satisfies More generally, we shall deal with homotopies between identity and other operators P, i.e., of the form In Euclidean space, the Laplacian provides us with such a homotopy. Write = dδ + δd. Denote by −1 the operator of convolution with the fundamental solution of the Laplacian. Then −1 commutes with d and its adjoint δ; hence K Euc = δ −1 satisfies I = d K Euc + K Euc d on globally defined L p differential forms. Furthermore, K Euc is bounded L p → L q provided 1 p − 1 q = 1 n . This proves the global Poincaré p,q (h) inequality for Euclidean space.
Rumin defines a Laplacian c by c = d c δ c + δ c d c when both d c and δ c are first-order horizontal differential operators, and by c = (d c δ c ) 2 + δ c d c or c = d c δ c + (δ c d c ) 2 near middle dimension (i.e., when h = n or h = n + 1, respectively), when one of them has order 2. This leads to a homotopy of the form K 0 = δ c −1 c depending on the degree. Again, K 0 is a singular integral of potential type associated with a homogeneous kernel and therefore is bounded from L p to L q under assumption E(h, p, q, n) (see [20] or [21] for the continuity of Riesz potentials in homogeneous groups). This proves the global H-Poincaré p,q (h) inequality for Heisenberg group, Theorem 1.1.

Local homotopy operators
We pass to interior estimates. In Euclidean space, Poincaré's lemma asserts that every closed form on a ball is exact. We need a quantitative version of this statement. The standard proof of Poincaré's lemma relies on a homotopy operator, which depends on the choice of an origin. Averaging over origins yields a bounded operator K Euc : L p → L q , as was observed by Iwaniec and Lutoborski [33]. This proves the global Euclidean Poincaré p,q (h) inequality for convex Euclidean domains. A support preserving variant J Euc : L p → L q appears in Mitrea et al. [42], and this proves the global Euclidean Sobolev p,q inequality for bounded convex Euclidean domains. Incidentally, since for balls, constants do not depend on the radius of the ball, this reproves the global Euclidean Sobolev p,q inequality for Euclidean spaces.
In this paper, a sub-Riemannian counterpart is obtained using the homotopy equivalence of the de Rham and Rumin complexes. Since this homotopy is a differential operator, a preliminary smoothing operation is needed. This is obtained by localizing (multiplying the kernel with cut-offs) the global homotopy K 0 provided by the inverse of Rumin's (modified) Laplacian.
Hence the proof goes as follows (see § 5): (1) Show that the inverse K 0 of Rumin's modified Laplacian on all of H n is given by a homogeneous kernel k 0 . Deduce bounds L p → W 1,q , where q, p are as above. Conclude that K 0 is an exact homotopy for globally defined L p forms. Basically, this step does not contain any new idea, relying only on the estimates of the fundamental solution of Rumin's modified Laplacian (see [10]) and on classical estimates for convolution kernels in homogeneous groups (see [20,21]).
(2) Take a smooth cut-off function ψ, ψ ≡ 1 in a neighborhood of the origin, and split k 0 = ψk 0 + (1 − ψ)k 0 , so that ψk 0 has small support near the origin and (1 − ψ)k 0 is smooth. Denote by T the convolution operator associated with the kernel ψk 0 , and by K smooth the convolution operator associated with the kernel (1 − ψ)k 0 . It turns out that T is a homotopy on balls (with a loss on domain) between the identity I and the operator S := d c K smooth + K smooth d c (which is smoothing), i.e., The operator S provides the required local smoothing operator. (

Global smoothing
Now we piece together local homotopy operators into globally defined smoothing operators. Let k 3. Let (M, H, g) be a bounded C k -geometry sub-Riemannian contact manifold. Pick a uniform covering by equal radius balls. Let χ j be a partition of unity subordinate to this covering. Let φ j be the corresponding charts from the unit Heisenberg ball. Let S j and T j denote the smoothing and homotopy operators associated with φ j using the pull-back operator. Set When d c is first order, the commutator [χ j , d c ] is an order 0 differential operator; hence T j [χ j , d c ] gains one derivative. When d c is second order, [χ j , d c ] is a first-order differential operator. It turns out that precisely in this case, T j gains two derivatives. Hence T j [χ j , d c ] gains one derivative in this case as well.
The details are discussed in § 6.

Structure of the paper
In § 2, we collect basic results about Heisenberg groups H n and differential forms in H n . Successively, we recall the notion of the Rumin complex for Heisenberg groups as well as for general contact manifolds, providing explicit examples in low dimensions. In § 3, we present a list of general results for Folland-Stein homogeneous kernels and, in particular, for matrix-valued kernels associated with Rumin's homogeneous Laplacian in H n . Section 4 is devoted to the theory of Folland-Stein Sobolev spaces in Heisenberg groups and in sub-Riemannian contact manifolds with bounded geometry. In particular, in § 4.2, we precisely provide the notion and the properties of manifolds with bounded geometry. Section 5 is the core of the paper, containing an approximate homotopy formula (i.e., a homotopy formula with a smoothing error term) and Poincaré and Sobolev inequalities for differential forms in H n . Then, in § 6, we are able to prove a similar approximate homotopy formula for sub-Riemannian contact manifolds with bounded geometry. The error term is a regularizing operator with 'maximal regularity'. Finally, § 7 contains a few examples of contact manifolds with bounded geometry and a brief discussion of the q, p cohomology.

Differential forms on Heisenberg groups
We denote by H n the (2n + 1)-dimensional Heisenberg group, identified with R 2n+1 through exponential coordinates. A point p ∈ H n is denoted by p = (x, y, t), with both x, y ∈ R n and t ∈ R. If p and p ∈ H n , the group operation is defined by Note that H n can be equivalently identified with C × R endowed with the group operation (z, t) · (ζ, τ ) := (z + ζ, t + τ − 1 2 I m (zζ )). The unit element of H n is the origin, which will be denoted by e. For any q ∈ H n , the (left) translation τ q : H n → H n is defined as For a general review on Heisenberg groups and their properties, we refer the reader to [30,56] and to [57]. We limit ourselves to fixing some notations, following [27].
First, we note that Heisenberg groups are smooth manifolds (and therefore are Lie groups). In particular, the pull-back of differential forms is well defined as follows (see, e.g., [28, Proposition 1.106]). Definition 2.1. If U, V are open subsets of H n , and f : U → V is a diffeomorphism, then for any differential form α of degree h, we denote by f α the pull-back form in U defined by for any h-tuple (v 1 , . . . , v h ) of tangent vectors at p.
The Heisenberg group H n can be endowed with the homogeneous norm (Cygan-Korányi norm): if p = (x, y, t) ∈ H n , then we set and we define the gauge distance (a true distance, see [56, p. 638], with a different normalization in the group law, which is left-invariant, i.e., d(τ q p, τ q p ) = d( p, p ) for all p, p ∈ H n ) as d( p, q) := ( p −1 · q).
Note that d is equivalent to the Carnot-Carathéodory distance on H n (see, e.g., [14,Corollary 5.1.5]). Finally, the balls for the metric d are the so-called Cygan-Korányi balls B( p, r ) := {q ∈ H n ; d( p, q) < r }.
Note that Cygan-Korányi balls are convex smooth sets. A straightforward computation shows that, if ρ( p) < 1, then It is well known that the topological dimension of H n is 2n + 1, since as a smooth manifold it coincides with R 2n+1 , whereas the Hausdorff dimension of (H n , d) is Q := 2n + 2 (the so-called homogeneous dimension of H n ).
We denote by h the Lie algebra of the left-invariant vector fields of H n . The standard basis of h is given, for i = 1, . . . , n, by The only non-trivial commutation relations are [X j , Y j ] = T , for j = 1, . . . , n. The horizontal subspace h 1 is the subspace of h spanned by X 1 , . . . , X n and Y 1 , . . . , Y n : h 1 := span {X 1 , . . . , X n , Y 1 , . . . , Y n }. Coherently, from now on, we refer to X 1 , . . . , X n , Y 1 , . . . , Y n (identified with first-order differential operators) as the horizontal derivatives. Denoting by h 2 the linear span of T , the two-step stratification of h is expressed by The stratification of the Lie algebra h induces a family of non-isotropic dilations δ λ : H n → H n , λ > 0 as follows: if p = (x, y, t) ∈ H n , then δ λ (x, y, t) = (λx, λy, λ 2 t).
The vector space h can be endowed with an inner product, indicated by ·, · , making X 1 , . . . , X n , Y 1 , . . . , Y n and T orthonormal.
Throughout this paper, we write also W i := X i , W i+n := Y i and W 2n+1 := T, for i = 1, . . . , n.
We put 0 h := 0 h = R and, for 1 h 2n + 1, In the sequel, we shall denote by h the basis of h h defined by To avoid cumbersome notations, if I := (i 1 , . . . , i h ), we write The inner product ·, · on 1 h yields naturally an inner product ·, · on h h making h an orthonormal basis. The volume (2n + 1)-form ω 1 ∧ · · · ∧ ω 2n+1 will be also written as d V .
Throughout this paper, the elements of h h are identified with left-invariant differential forms of degree h on H n .
The same construction can be performed starting from the vector subspace h 1 ⊂ h, obtaining the horizontal h-covectors It is easy to see that provides an orthonormal basis of h h 1 . Keeping in mind that the Lie algebra h can be identified with the tangent space to H n at x = e (see, e.g., [28, Proposition 1.72]), starting from h h we can define by left translation a fiber bundle over H n that we can still denote by h h. We can think of h-forms as sections of h h. We denote by h the vector space of all smooth h-forms.
We already pointed out in § 1.2 that the stratification of the Lie algebra h yields a lack of homogeneity of de Rham's exterior differential with respect to group dilations δ λ . Thus, to take into account the different degrees of homogeneity of the covectors when they vanish on different layers of the stratification, we introduce the notion of weight of a covector as follows.
We stress that generic covectors may fail to have a pure weight: it is enough to consider H 1 and the covector d x 1 + θ ∈ 1 h. However, the following result holds (see [8, formula (16) where h, p h denotes the linear span of the h-covectors of weight p. By our previous remark, decomposition (12) is orthogonal. In addition, since the elements of the basis h have pure weights, a basis of h, p h is given by h, p := h ∩ h, p h (such a basis is usually called an adapted basis).
We note that, according to (12), the weight of an h-form is either h or h + 1 and there are no forms of weight h + 2 since there is only one 1-form of weight 2. Something analogous can be possible for instance in H n × R, but it fails to be possible already in the case of general step 2 groups with higher-dimensional center (see also Remark 5.22).
As above, starting from h, p h, we can define by left translation a fiber bundle over H n that we can still denote by h, p h. Thus, if we denote by h, p the vector space of all smooth h-forms in H n of weight p, i.e., the space of all smooth sections of h, p h, we have

The Rumin complex on Heisenberg groups
Let us give a short introduction to the Rumin complex. For a more detailed presentation, we refer the reader to Rumin's papers [53]. Here we follow the presentation of [8]. The exterior differential d does not preserve weights. It splits into where d 0 preserves weight, d 1 increases weight by 1 unit and d 2 increases weight by 2 units. More explicitly, let α ∈ h be a (say) smooth form of pure weight h. We can write It is crucial to note that d 0 is an algebraic operator in the sense that for any real-valued so that its action can be identified at any point with the action of a linear operator from h h to h+1 h (which we denote again by d 0 ).
As above, E • 0 defines by left translation a fiber bundle over H n , which we still denote by E • 0 . To avoid cumbersome notations, we denote also by E • 0 the space of sections of this fiber bundle.
Let L : h h → h+2 h be the Lefschetz operator defined by Then the spaces E • 0 can be defined explicitly as follows.
Theorem 2.5 (See [50,52]). We have the following: (v) if * denotes the Hodge duality associated with the inner product in • h and the volume form d V , then Note that all forms in E h 0 have weight h if 1 h n and weight h + 1 if n < h 2n + 1. A further geometric interpretation (in terms of decomposition of h and of graphs within H n ) can be found in [26].
Note that there exists a left-invariant orthonormal basis of E h 0 that is adapted to filtration (12). Such a basis is explicitly constructed by induction in [1].
The core of Rumin's theory consists in the construction of a suitable 'exterior differential' d c : Let us sketch Rumin's construction: first, the next result (see [8,Lemma 2.11] for a proof) allows us to define a (pseudo) inverse of d 0 : With the notations of the previous lemma, we set We note that d −1 0 preserves the weights. The following theorem summarizes the construction of the intrinsic differential d c (for details, see [53] and [8, Section 2]).
Let E be the projection on E along F (which is not an orthogonal projection). We have the following: Set now We have the following: 0 is a homogeneous differential operator in the horizontal derivatives of order 1 if h = n, whereas d c : E n 0 → E n+1 0 is a homogeneous differential operator in the horizontal derivatives of order 2.
To illustrate the previous construction, let us write explicitly the classes E h 0 and the differential d c : in H 1 and H 2 (for proofs, see e.g., [2]).
Example 2.8. Consider the first Heisenberg group H 1 ≡ R 3 with variables (x, y, t). With the notations of (11), we have . In this case, The classes E 3 0 and E 4 0 are easily written by Hodge duality: Remark 2.10. The construction of the Rumin complex can be carried out in general Carnot groups following verbatim the construction presented in § 2.2 for Heisenberg groups, once a general notion of weight is provided. This can be easily done in terms of homogeneity of a covector with respect to group dilations (see [8,53,54]).
Since the exterior differential d c on E h 0 can be written in coordinates as a left-invariant homogeneous differential operator in the horizontal variables, of order 1 if h = n and of order 2 if h = n, the proof of the following Leibniz formula is easy.
Lemma 2.11. If ζ is a smooth real function, then we have the following: is a linear homogeneous differential operator of degree zero, with coefficients depending only on the horizontal derivatives of ζ .
where P n 1 : is a linear homogeneous differential operator of degree 1, with coefficients depending only on the horizontal derivatives of ζ , and where P h 0 : E n 0 → E n+1 0 is a linear homogeneous differential operator in the horizontal derivatives of degree 0 with coefficients depending only on second-order horizontal derivatives of ζ .
The next remarkable property of the Rumin complex is its invariance under contact transformations. Here we state a special case before developing this point in § 2.3 (see [9,Proposition 3.19] for a proof).
Proposition 2.12. If we write a form α = j α j ξ h j in coordinates with respect to a left-invariant basis of E h 0 (see (15)), we have for all q ∈ H n . In addition, for t > 0, and

The Rumin complex in contact manifolds
Let us start with the following definition (see [41,).
We recall that, by a classical theorem of Darboux, any contact manifold (M, H ) is locally contact diffeomorphic to the Heisenberg group H n (see [41, p. 112]).
Rumin's intrinsic complex is invariantly defined for general contact manifolds (M, H ). Although the operators d 0 and d −1 0 are not invariantly defined, the subspaces E and F of differential forms, the operator E onto E parallel to F, the vector bundles E h 0 and the projector E 0 are contact invariants. To see this, let us follow [54].
Let us start with E and F. It is well known that, for every h n − 1, Changing θ to another smooth 1-form θ = f θ with kernel H does not change V and W . With these choices, spaces of smooth sections of V and W (which we still denote by V and W ) depend only on the plane field H . We can define subspaces of smooth differential . We note that for a differential form ω such that ω |H ∈ E 0 , E (ω) only depends on ω |H . It follows that ( E ) |E 0 can be viewed as defined on the space of sections of E 0 (still denoted by E 0 ), which is a contact invariant. Since and d c viewed as an operator on E 0 , is a contact invariant. In the sequel, we shall ignore the distinction between E 0 and E 0 . We shall denote E • 0 = h E h 0 endowed with the exterior differential d c . Alternative contact invariant descriptions of the Rumin complex can be found in [13,16].
By construction, we have the following: 0 is a homogeneous differential operator in the horizontal derivatives of order 1 if h = n, whereas d c : E n 0 → E n+1 0 is a homogeneous differential operator in the horizontal derivatives of order 2.
The following statement expresses the fact that the Rumin complex is invariant under contactomorphism. In other words, the pull-back map is natural, i.e., it is a chain map for (E • 0 , d c ).
Proposition 2.14. If φ is a contactomorphism from an open set U ⊂ H n to M, and we denote by V the open set V := φ(U), the pull-back operator φ # satisfies the following: , h = n and a differential operator of order 1 if v ∈ E n 0 (U). Proof. Assertions (i) and (ii) follow straightforwardly since φ is a contact map. Assertion (iii) follows from Lemma 2.11 since, by definition,

Kernels in Heisenberg groups
Following a classical notation [55] Following e.g., [21, p. 15], we can define a group convolution in H n : if, for instance, f ∈ D(H n ) and g ∈ L 1 loc (H n ), we set We recall that if (say) g is a smooth function and P is a left-invariant differential operator, then We recall also that the convolution is again well defined when f, g ∈ D (H n ), provided at least one of them has compact support. In this case, the following identities hold, for any test function φ.
As in [21], we also adopt the following multi-index notation for higher-order derivatives.
By the Poincaré-Birkhoff-Witt theorem, the differential operators W I form a basis for the algebra of left-invariant differential operators in H n . Furthermore, we set |I | := i 1 + · · · + i 2n + i 2n+1 the order of the differential operator W I , and d(I ) := i 1 + · · · + i 2n + 2i 2n+1 its degree of homogeneity with respect to group dilations.
Definition 3.1. Let ω ∈ D(H n ) be supported in the unit ball B(e, 1), and assume ω(x) d x = 1. If ε > 0, we denote by ω ε the Friedrichs mollifier ω ε (x) := ε −Q ω(δ 1/ε x). The procedure of regularization by convolution can be extended componentwise to differential forms in L 1 loc (H n , E • 0 ), as follows: if α = j α j ξ h j , we set As above, denote by * the group convolution in H n . By [21] and (23), if u ∈ L 1 loc (H n ), the convolution u ε := u * ω ε enjoys the same properties of the usual regularizing convolutions in Euclidean spaces.
Following [20], we recall now the notion of kernel of type µ.
Definition 3.2. A kernel of type µ is a homogeneous distribution of degree µ − Q (with respect to group dilations δ r ), which is smooth outside of the origin. The convolution operator with a kernel of type µ is called an operator of type µ. (i) v K is again a kernel of type µ.
(ii) W K and K W are associated with kernels of type µ − 1 for any horizontal derivative W .
Theorem 3.4. Suppose 0 < α < Q, and let K be a kernel of type α. Then we have the following: for all u ∈ L p (H n ). for all u ∈ L p (H n ) with supp u ⊂ B.
(iii) If K is a kernel of type 0 and 1 < p < ∞, then there exists C = C( p) > 0 such that u * K L p (H n ) C u L p (H n ) .
Proof. For statements (i) and (iii), we refer the reader to [20, Propositions 1.11 and 1.9]. As for (ii), if p Q/α, we choose 1 <p < Q/α such that 1/p 1/q + α/Q. If we set 1/q := 1/p − α/Q < 1/q, then Lemma 3.5. Suppose 0 < α < Q. If K is a kernel of type α and ψ ∈ D(H n ), ψ ≡ 1, in a neighborhood of the origin, then statements (i) and (ii) of Theorem 3.4 still hold if we replace K by (1 − ψ)K . Analogously, if K is a kernel of type 0 and ψ ∈ D(H n ), then statement (iii) of Theorem 3.4 still holds if we replace K by (ψ − 1)K .
The following estimate will be useful in the sequel. In addition, let g be a smooth function in H n \ {0} satisfying the logarithmic estimate |g( p)| C(1 + | ln | p||), and suppose its horizontal derivatives are kernels of type Q − 1 with respect to group dilations. Then, if f ∈ D(H n ) and R is a homogeneous polynomial of degree 0 in the horizontal derivatives, we have Proof. The first part of the lemma is a particular instance of [21,Lemma 6.4]. As for the second part, we can repeat the same argument. Indeed, the first statement follows straightforwardly from the first part of the lemma since, by the Poincaré-Birkhoff-Witt theorem, we can write where the differential operators R j have homogeneous degree − 1. Finally, the last statement can be proved as follows: suppose supp f ⊂ B(0, M), M > 1, and take | p| > 2M. Then, keeping in mind that 1 < M |q −1 p| M + | p|,

Rumin's Laplacians
In this section, we recall the main properties of Rumin's generalization of the Laplace operator in Heisenberg groups. In order to introduce this operator, we need preliminarily the following property about the L 2 -adjoint of Rumin's exterior differential d c .
Definition 3.9. In H n , following [50], we define the operator H,h on E h 0 by setting For the sake of simplicity, since a basis of E h 0 is fixed, any α ∈ E h 0 can be identified with the vector (α 1 , . . . , α N h ) of its components, and the operator H,h can be identified with a matrix-valued map, still denoted by H,h , where N h is the dimension of E h 0 (N h is explicit in Theorem 2.5, (iv)) and D (H n , R N h ) is the space of vector-valued distributions on H n . This identification makes it possible to avoid the notion of currents. We refer the reader to [8] for a more elegant presentation.  It is proved in [50] that H,h is hypoelliptic and maximal hypoelliptic in the sense of [32]. In general, if L is a differential operator on D (H n , R N h ), then L is said to be hypoelliptic if for any open set V ⊂ H n where Lα is smooth, then α is smooth in V. In addition, if L is homogeneous of degree a ∈ N, we say that L is maximal hypoelliptic if for any δ > 0, there exists C = C(δ) > 0 such that for any homogeneous polynomial P in W 1 , . . . , W 2n of degree a, we have The next theorem provides a key tool for the present paper: the existence of a suitable 'inverse' −1 H,h of H,h that is associated with a vector-valued kernel, which we still denote by −1 H,h . Combining [50, Section 3] and [10, Theorems 3.1 and 4.1], we obtain the following result. We stress again the fact that the order of H,h with respect to group dilations is 2 if h = n, n + 1 whereas it is 4 if h = n, n + 1.
with the following properties: (i) If a < Q, then the K i j 's are kernels of type a, for i, j = 1, . . . , N h . If a = Q, then the K i j 's satisfy the logarithmic estimate |K i j ( p)| C(1 + | ln ρ( p)|) and hence belong to L 1 loc (H n ). Moreover, their horizontal derivatives W K i j , = 1, . . . , 2n, are kernels of type Q − 1. then Remark 3.13. Coherently with formula (24), the matrix-valued operator −1 H,h can be identified with an operator (still denoted by −1 H,h ) acting on smooth compactly supported differential forms in D(H n , E h 0 ). Moreover, when the notation will not be misleading, we shall denote by −1 H,h its kernel.
The following lemma states that d c and H commute.
Lemma 3.14. We note that the Laplace operator commutes with the exterior differential d c . More precisely, if α ∈ C ∞ (H n , E h 0 ) and n 1, Proof. The proof is an easy consequence of the fact that d 2 c = 0. Indeed, let us prove (i).
An analogous argument applies to (iii) and (iv).
The commutation of d c and δ c with −1 H follows from the previous lemma.
We are left then with the proof of claim (28).
Suppose first h = n − 1, n, n + 1. By Lemma 3.14(i) and by Theorem 3.12, we have If h = n − 1, then by Lemma 3.14(iii), (iv) and by Theorem 3.12, H,n α is a form of degree n + 1 and −1 H,n α is a form of degree n, we use again Lemma 3.14(i)) Finally, if h = n + 1, then again from Lemma 3.14(i), This proves (28), and hence we have proved (i)-(iii). Since δ c = (−1) h * d c * , and keeping in mind Remark 3.11, the remaining assertions (iv)-(vi) follow by the Hodge duality from (i)-(iii).

Sobolev spaces on Heisenberg groups
Let U ⊂ H n be an open set and let 1 p ∞ and m ∈ N; W m, p Euc (U ) denotes the usual Sobolev space. We want now to introduce intrinsic (horizontal) Sobolev spaces.
Since here we are dealing only with integer order Folland-Stein function spaces, we can give this simpler definition (for a general presentation, see e.g., [20]). Folland-Stein Sobolev spaces enjoy the following properties akin to those of the usual Euclidean Sobolev spaces (see [20] and, e.g., [25]).
If U is bounded, then we can take f 0 = 0. Finally, we stress that This is nothing but the intuitive notion of 'currents as differential forms with distributional coefficients'. The action of u ∈ W −m, p (U, On the other hand, suppose for the sake of simplicity that U is bounded. Then by Remark 4.5, there exist f j I ∈ L p (U ), j = 1, . . . , N h , i = 1, . . . , 2n + 1 such that Alternatively, one can express duality in spaces of differential forms using the pairing between h-forms and (2n + 1 − h)-forms defined by Note that this makes sense for Rumin forms and is a non-degenerate pairing. In this manner, the dual of L p (U, ). Hence W −m, p (U, E h 0 ) consists of differential forms of degree 2n + 1 − h whose coefficients are distributions belonging to W −m, p (U ).
In the Riemannian setting, Sobolev spaces of differential forms are invariant with respect to the pull-back operator associated with sufficiently smooth diffeomorphisms (see, e.g., [55,Lemma 1.3.9]). An analogous statement holds for Folland-Stein Sobolev spaces in Heisenberg groups, provided we restrict ourselves to contact diffeomorphisms. Indeed, we have the following lemma. Then the pull-back operator φ from W , p forms on V to W , p forms on U is bounded, and its norm depends only on the C k norms of φ and φ −1 . Proof. When 0, this follows from the chain rule and the change of variables formula. According to the change of variables formula the adjoint of φ with respect to the above pairing is (φ −1 ) . Hence φ is bounded on negative Sobolev spaces of differential forms as well.

Sobolev spaces on contact sub-Riemannian manifolds with bounded geometry
First of all, let us give the definition of contact manifolds of bounded geometry. Definition 4.9. Let k be a positive integer and let B(e, 1) denote the unit sub-Riemannian ball in H n . We say that a sub-Riemannian contact manifold (M, H, g) has bounded C k -geometry if there exist constants r, C > 0 such that, for every x ∈ M, there exists a contactomorphism (i.e., a diffeomorphism preserving the contact forms) φ x : B(e, 1) → M that satisfies (1) B(x, r ) ⊂ φ x (B(e, 1)); (2) φ x is C-bi-Lipschitz, i.e., , φ x (q)) Cd( p, q) for all p, q ∈ B(e, 1);  More examples arise from covering spaces of such compact manifolds. Note that every orientable compact 3-manifold admits a contact structure [39], it can be equipped with sub-Riemannian structures and its universal covering space is usually non-compact. This leads to a large variety of non-compact bounded geometry sub-Riemannian contact 3-manifolds.
The following covering lemma is basically [40, Theorem 1.2]. Lemma 4.11. Let (M, H, g) be a bounded C k -geometry sub-Riemannian contact manifold, where k is a positive integer. Then there exist ρ > 0 (depending only on the radius r of Definition 4.9) and an at most countable covering {B(x j , ρ)} of M such that we have the following: (i) Each ball B(x j , ρ) is contained in the image of one of the contact charts of Definition 4.9. For any j ∈ N, let S j be a countable dense subset of φ −1 x j (V x j ); then φ x j (S j ) is a countable dense subset of V x j . Thus Σ := j φ x j (S j ) is a countable dense subset of M.
Finally, note that our previous arguments yield also that the covering is uniformly locally finite. Indeed, let x be fixed and let B( , and the assertion follows again by a doubling argument in H n . We can define now Sobolev spaces (involving a positive or negative number of derivatives) on bounded geometry contact sub-Riemannian manifolds.
Definition 4.12. Let k be a positive integer, let (M, H, g) be a bounded C k -geometry sub-Riemannian contact manifold, and let {χ j } be a partition of unity subordinate to the atlas U := {B(x j , ρ), φ x j } of Lemma 4.11. From now on, for the sake of simplicity, we shall write φ j := φ x j . We stress that φ −1 j (supp χ j ) ⊂ B(e, 1). If α is a Rumin differential form on M, we say that α ∈ W , p is compactly supported in B(e, 1) and therefore can be continued by zero on all of H n ). Then we set The following result shows that the definition of the Sobolev spaces W , p U (M, E • 0 ) does not depend on the atlas U. Therefore, once the proposition is proved, we drop the index U from the notation for Sobolev norms.
Proposition 4.13. Let k and be as above, and let (M, H, g) be a bounded C k -geometry sub-Riemannian contact manifold. If U := {B(y j , ρ ), φ y j } is another atlas of M satisfying Definition 4.9 and Lemma 4.11 with the same choice of ρ, and {χ j } is an associated partition of unity, then with equivalent norms.
Proof. Let j ∈ N be fixed, and let (B(x j , ρ), φ j ) be a chart of U. We can write where #I j N , since, by Lemma 4.11(iii), B(x j , ρ) is covered by at most N balls of the covering associated with U . Thus, by Definition 4.9(3) and keeping in mind that

Homotopy formulas and Poincaré and Sobolev inequalities
In this paper, we are mainly interested to obtain functional inequalities for differential forms that are the counterparts of the classical ( p, q)-Sobolev and Poincaré inequalities on a ball B ⊂ R n with sharp exponents of the form (as well as of its counterpart for compactly supported functions). In this case, we can choose q = pn/(n − p), provided p < n.  We give below a statement that deals with H-Sobolev inequality.
Definition 5.5. Take λ > 1 and set B = B(e, 1) and B = B(e, λ). If 1 h 2n, 1 p q < ∞ and q p, we say that the interior H-Sobolev p,q (h) inequality holds if there exists a constant C such that for every compactly supported smooth d c -exact differential h-form ω in L p (B; E h 0 ), there exists a smooth compactly supported differential Here we have extended ω by 0 to all of B .
Remark 5.6. If h = 1 and Q > p 1, then ( H-Sobolev p,q (1)) is nothing but the usual Sobolev inequality with In [33], starting from Cartan's homotopy formula, the authors proved that if D ⊂ R N is a convex set, 1 < p < ∞, 1 < h < N , then there exists a bounded linear map which is a homotopy operator, i.e., where ψ ∈ D(D), D ψ(y) dy = 1, and Starting from [33], in [42,Theorem 4.1], the authors define a compact homotopy operator J Euc,h in Lipschitz star-shaped domains in the Euclidean space R N , providing an explicit representation formula for J Euc,h , together with continuity properties among Sobolev spaces. More precisely, if D ⊂ R N is a star-shaped Lipschitz domain and 1 < h < N , then there exists Furthermore, J Euc,h maps smooth compactly supported forms to smooth compactly supported forms. Take now D = B(e, 1) =: B and N = 2n (for the sake of simplicity, from now on we drop the index k -the degree of the formwriting, e.g., K Euc instead of K Euc,hk ). Analogously, we can define Then K and J invert Rumin's differential d c on closed forms of the same degree. More precisely, we have the following lemma.
Lemma 5.7. If ω is a smooth d c -exact differential form, then In addition, if ω is compactly supported in B, then J ω is still compactly supported in B.
Proof. We prove for instance the identity for d c K ω. If d c ω = 0, then d( E ω) = 0, and hence E ω = d K Euc ( E ω), by (33). We recall now that by Theorem 2.7(ii) and (iv), d E = E d and both E E 0 E = E and E 0 E E 0 = E 0 . Thus, by (36), Finally, if supp ω ⊂ B, then supp J ω ⊂ B since both E and E 0 preserve the support. (i) if 1 < p < ∞ and h = 1, . . . , 2n + 1, then K : is bounded. Analogous assertions hold for 1 h 2n + 1 when we replace K by J . In addition, supp J ω ⊂ B.
Proof. By its very definition, E : . Then we can conclude the proof of (i), taking again into account that E is a differential operator of order 1 in the horizontal derivatives.
To prove (ii), it is enough to recall that K = E 0 K Euc on forms of degree h > n, together with [33, Remark 4.1].
As for (iii), the statement can be proved similarly to (i), noting that K = E 0 E K Euc on forms of degree n + 1.
Finally, supp J ω ⊂ B since both E and E 0 preserve the support.
The operators K and J provide a local homotopy in the Rumin complex, but fail to yield the Sobolev and Poincaré inequalities we are looking for since, because of the presence of the projection operator E (which on forms of low degree is a first-order differential operator), they lose regularity as is stated in Lemma 5.8 (ii). In order to build 'good' local homotopy operators with the desired gain of regularity, we have to combine them with homotopy operators which, though not local, in fact provide the 'good' gain of regularity. Proposition 5.9. If α ∈ D(H n , E h 0 ) for p > 1 and h = 1, . . . , 2n, then the following homotopy formulas hold: there exist operators K 1 ,K 1 and K 2 ,K 1 acting on D(H n , E • 0 ) such that • if h = n, n + 1, then α = d c K 1 α +K 1 d c α, where K 1 andK 1 are associated with kernels k 1 ,k 1 of type 1; • if h = n, then α = d c K 1 α +K 2 d c α, where K 1 andK 2 are associated with kernels k 1 ,k 2 of types 1 and 2, respectively; • if h = n + 1, then α = d c K 2 α +K 1 d c α, where K 2 andK 1 are associated with kernels k 2 ,k 1 of types 2 and 1, respectively.
Example 5.11. Suppose for instance n = 1. In this case Q = 4 and, keeping in mind Example 2.8, α = α 1 d x + α 2 dy ∈ E 1 0 , then Corollary 5.10 yields that there exists a function φ such that Moreover, if α = α 13 d x ∧ θ + α 23 dy ∧ θ ∈ E 2 0 , then there exists φ = φ 1 d x + φ 2 dy ∈ E 1 0 such that Theorem 5.12. Let B = B(e, 1) and B = B(e, λ), λ > 1, be concentric balls of H n . If Proof. Suppose first h = n, n + 1. We consider a cut-off function ψ R supported in an R-neighborhood of the origin such that ψ R ≡ 1 near the origin. With the notations of Proposition 5.9, we can write where are the matrix-valued kernels associated with the operators δ c H,h and δ c H,h+1 , respectively, as shown in the proof of Proposition 5.9. Let us denote by K 1,R ,K 1,R the convolution operators associated with ψ R k 1 , ψ Rk1 , respectively. Let us fix two balls B 0 , B 1 with and a cut-off function We have We set T α := K 1,R α 0 ,T d c α :=K 1,R d c α 0 , Sα := S 0 α 0 .
We note that, provided R > 0 is small enough, the definition of T andT does not depend on the continuation of α outside B 0 . By (44), we have If h = n, we can carry out the same construction, replacingk 1 byk 2 (keep in mind that k 2 is a kernel of type 2, again by Proposition 5.9). Analogously, if h = n + 1, we can carry out the same construction, replacing k 1 by k 2 (again a kernel of type 2).
Later on, we need the following remark. The homotopies T andT provide the desired 'gain of regularity' as stated in the following theorem.
Theorem 5.14. Let B = B(e, 1) and B = B(e, λ), λ > 1, be concentric balls of H n , and let 1 h 2n + 1. If T,T are as in Theorem 5.12, then we have the following: . In addition, for every (h, p, q) satisfying inequalities we have Let us prove (i). Suppose h = n, and take β ∈ W −1, p (B , E h+1 0 ). As in the proof of the previous theorem, let ψ R be a cut-off function supported in an R-neighborhood of the origin such that ψ R ≡ 1 near the origin. Thus, again with the notations of the proof of the previous theorem (see, in particular, (46) and (42)), the operatorK 1,R is associated with a matrix-valued kernel ψ R (k 1 ) ,λ and β is identified with a vector-valued distribution (β 1 , . . . , β N h ), with β j = i W i f j i as in Remark 4.7, with As in the proof of the previous theorem, let us fix two balls B 0 , B 1 with B B 0 B 1 B . If χ ∈ D(B 1 ) is a cut-off function such that χ ≡ 1 on B 0 , we set β 0 = χβ. Thus (β 0 ) j , the jth component of β 0 , has the form Keeping in mind Remark 4.7, in order to estimate the norm ofT β in By (29), T β | φ is a sum of terms of the form (where, as above, f 0 = χ f ) or of the form where κ denotes one of the kernels (k 1 ) ,λ of type 1 associated withk 1 (see (41) in the proof of the previous theorem), f is one of the f j i 's and φ is one of the φ j 's. As for (48), by (23), We note now that v W I v κ is a kernel of type 0. Therefore, by Lemma 3.5 (keep in mind that f 0 and φ are real functions), . The term in (49) can be handled in the same way, taking into account Remark (3.6). Eventually, combining (48) and (49), we obtain that . The assertion for h = n can be proved in the same way, taking into account that T is built from a kernel of type 2 and that the norm in the space W −2, p (B, E n+1 0 ) is expressed by duality in terms of second-order horizontal derivatives of test functions (see Remark 4.5).
Let us prove now (ii). Suppose h = n + 1 and take α = j α j ξ h j ∈ D(B , E h 0 ). Arguing as above, in order to estimate T α W 1, p (B,E h−1 0 ) , we have to consider terms of the form W (ψ R κ * (χ α j )) = ψ R κ * (W (χ α j )) (when we want to estimate the L p -norm of the horizontal derivatives of T α) or of the form (when we want to estimate the L p -norm of T α). Both (50) and (51) can be handled as in case (i) (no need here of the duality argument). We point out that (51) yields an L p − L q estimate (since, unlike (50), it involves only kernels of type 1) and then assertion (iii) follows.
The operator S is the required local smoothing operator. More precisely, we have the following theorem.  1) and B = B(e, λ), λ > 1, be concentric balls of H n , and let 1 h 2n + 1. Then the operator S defined in (46) is a smoothing operator. In particular, for any m, s ∈ Z, m < s, S is bounded from W m, p (B , E h 0 ) to W s,q (B, E h 0 ) for any p, q ∈ (1, ∞) and maps W m, p (B , E h 0 ) into C ∞ (B, E h 0 ). Proof. Since B is bounded, we can assume q > p. First, take m = 0. Again, let us fix two balls B 0 , B 1 with B B 0 B 1 B . If χ ∈ D(B 1 ) is a cut-off function such that χ ≡ 1 on B 0 , we set α 0 = χ α. Keeping the notations of the proof of Theorem 5.12, it is easy to check that Sα can be written as (see (45)) Thus, if α = j α j ξ h j , then each entry of Sα is a sum of terms of the form where κ is a smooth kernel. Thus we are led to estimate the L q -norms in B of a sum of terms of the form and the assertion follows by the classical Hausdorff-Young inequality (see [20,Proposition 1.10]) since the kernel 1 2B W J κ belongs to all L r , r 1. Therefore S is bounded from . Clearly, this yields the continuity of S from W m, p (B , E h 0 ) to W s,q (B, E h 0 ) for m 0. The proof in the case m < 0 can be carried out by a duality argument akin to the one we used in the proof of Theorem 5.14.
Remark 5.16. Apparently, in the previous theorem, two different homotopy operators T andT appear. In fact, they coincide when acting on the form of the same degree.
More precisely, in Proposition 5.9, the homotopy formulas involve four operators K 1 ,K 1 , K 2 ,K 2 , where the notation is meant to distinguish operators acting on d c α (the operators with tilde) from those on which the differential acts (the operators without tilde), whereas the lower index 1 or 2 denotes the type of the associated kernels. Alternatively, a different notation could be used: if α ∈ D(H n , E h 0 ), we can write where the tilde has the same previous meaning, whereas the lower index refers now to the degree of the forms on which the operator acts.
It is important to note that H,h+1 (as it appears in the homotopy formula at the degree h), which equals K h+1 (as it appears in the homotopy formula at the degree h + 1 n − 1). Take now h = n − 1. ThenK n = δ c d c δ c −1 H,n (as it appears in the homotopy formula at the degree n), which equals K n (as it appears in the homotopy formula at the degree n). If h = n, thenK n+1 = δ c −1 H,n+1 (as it appears in the homotopy formula at the degree n), which equals K n+1 (as it appears in the homotopy formula at the degree n + 1). Finally, if h > n, thenK h+1 = δ c −1 H,h+1 (as it appears in the homotopy formula at the degree h), which equals K h+1 (as it appears in the homotopy formula at the degree h + 1).
Once this point is established, from now on, we shall write Therefore T =T , and the homotopy formula (40) reads as It is worth pointing out the following fact.
Remark 5.17. As above, take B := B(e, 1), B := B(e, λ) with λ > 1, and as in formula (43), let us fix two balls B 0 , B 1 with and a cut-off function The following commutation lemma will be helpful in the sequel.
Indeed, take α ∈ L p (B , E h 0 ), and let χ 1 be a cut-off function supported in B , χ 1 ≡ 1 on B 1 . By convolution with the usual Friedrichs mollifiers (see Definition 3.1), we can find a sequence (α k ) k∈N in D(B , E h 0 ) converging to χ 1 α in L p (B , E h 0 ). By Theorem 5.14, On the other hand, χ 1 α ≡ α in B 1 , and then by Remark 5.17, S( , and hence by Theorem 5.12, Sd c α k → Sd c (χ 1 α) in B as k → ∞. Again, d c (χ 1 α) ≡ d c α in B 1 , and then by Remark 5.17, Sd c α k → Sd c α in B as k → ∞.
Finally, since d c Sα k = Sd c α k for all k ∈ N, we can take the limits as k → ∞ and the assertion follows.
The following theorem contains one of the main results of the paper: it yields the interior Poincaré inequality and Sobolev inequality for Rumin forms in the sense of Definitions 5.1 and 5.5.
Theorem 5.19. Take λ > 1 and set B = B(e, 1) and B = B(e, λ). If 1 h 2n + 1, as in (47), take Then we have the following: (i) An interior H-Poincaré p,q (h) inequality holds with respect to the balls B and B .
(ii) In addition, an interior H-Sobolev p,q (h) inequality holds for 1 h 2n.
Let ω ∈ L p (B, E h 0 ) be a compactly supported form such that d c ω = 0. Since ω vanishes in a neighborhood of ∂ B, without loss of generality, we can assume that it is continued by zero on B . In addition, ω = χ ω since χ ≡ 1 on supp ω.
By (53), we have ω = d c T ω + Sω. On the other hand, since ω vanishes outside B, by its very definition (see (46)), T ω is supported in B 0 by Remark 5.13 so that also Sω is supported in B 0 .
Again as above, Sω ∈ C ∞ (B, E h 0 ) and d c Sω = 0. Thus we can apply (38) to Sω and we get Sω = d c J Sω, where J is defined in (37) (that preserves the support). By Lemma 5.7, J Sω is supported in B 0 ⊂ B . Thus, if we set φ := (J S + T )ω, then φ is supported in B . Moreover, d c φ = d c K Sω + d c T ω = Sω + ω − Sω = ω.
At this point, we can repeat estimates (57) and we get eventually . This completes the proof of the theorem.
If p ∈ H n and t > 0, then the map x → f (x) := τ p δ t (x) maps B(e, ρ) into B( p, tρ) for ρ > 0. Therefore, by Proposition 2.12, from the previous theorem for balls of fixed radius, we obtain the following result for general balls.
Theorem 5.20. Take 1 h 2n + 1. Suppose 1 < p < Q if h = n + 1 and 1 < p < Q/2 if h = n + 1. Let q p such that Then there exists a constant C such that for every d c -closed differential h-form ω in and Analogously, if 1 h 2n, there exists a constant C such that for every compactly supported d c -closed h-form ω in L p (B( p, r ); E h 0 ), there exists a compactly supported Proof. We just have to take the pull-back f # ω and then apply Theorem 5.19.
If the choice of q is sharp (i.e., in (58), the equality holds), then the constant on the right-hand side of (61) is independent of the radius of the ball so that a global H-Sobolev p,q (h) inequality holds.
Therefore we get the following result.
In the case H 1 , for 1-forms and 2-forms for instance, the primitive φ of a compactly supported form can be written explicitly as in Example 5.11.
Remark 5.22. A scaling argument shows easily that the exponents in (59) and (60) are sharp. On the other hand, we have already discussed in § 1.5 whether similar sharp results can be proved for general Carnot groups, stating ultimately that this is not possible (at least relying on our present arguments). Now the argument of § 1.5 can be made more precise. If we look at the proofs of our inequalities, we see that at the very beginning, there is an approximate homotopy formula that in turn descends from the existence of a fundamental solution for a suitable hypoelliptic homogeneous 'artificial Laplacian'. This construction is still possible in general Carnot groups (see [8,48]) relying on the construction of a '0-order Laplacian', but the approximate homotopy formula involves singular integral operators that fail to have the good homogeneity. This is due to the fact that in general Carnot groups, with the exception of very particular cases, the forms of a given degree in the Rumin complex have different weights (this does not happen in Euclidean spaces and in Heisenberg groups). We stress that this phenomenon appears already in step 2 groups, very akin to Heisenberg groups, like quaternionic Heisenberg groups (see [11]), which are defined by replacing the complex field C by the field of quaternions in the definition of H 1 . This generates a two-step Carnot group whose center is 3-dimensional (while the center in H n is 1-dimensional).

Contact manifolds and global smoothing
Throughout this section, (M, H, g) will be a sub-Riemannian contact manifold of bounded C k -geometry as in Definition 4.9, k 3. We shall denote by (E • 0 , d c ) both the Rumin complex in (M, H, g) and in the Heisenberg group.
The core of this section consists in the proof of an approximate homotopy formula where the 'error term' S M has the maximal regularizing property compatible with the regularity of M, and T M enjoys the natural continuity properties between Sobolev spaces on M. The proof will be carried out in two steps: first (Lemma 6.1), we shall prove an approximate homotopy formula akin to (63), where S M 'gains only one horizontal derivative', and then iterating (63), we obtain the desired approximate homotopy formula, where S M has the maximal regularizing property compatible with the regularity of M. As in Definition 4.12, let now {χ j } be a partition of the unity subordinate to the atlas U := {B(x j , ρ), φ x j } of Lemma 4.11. From now on, for the sake of simplicity, we shall write φ j := φ x j . We stress again that φ −1 j (supp χ j ) ⊂ B(e, 1).
We can write We use now the homotopy formula in H n (see Theorem 5.12): Without loss of generality, we can assume that R > 0 in the definition of the kernel of T has been chosen in such a way that the R-neighborhood of φ −1 j (supp χ j ) ⊂ B(e, 1). In particular, v j − d c T v j − T d c v j is supported in B(e, 1) and therefore also Sv j is supported in B(e, 1). , 1)) so that it can be continued by zero on M. Thus

We set
and Lemma 6.1. Let (M, H, g) be a bounded C k -geometry sub-Riemannian contact manifold with k 3. If 2 k − 1 and T and S are defined in (64) and (65), then the following homotopy formula holds: In particular, Sd c = d c S. In addition, if 1 h 2n + 1, the following maps are continuous: First of all, we note that if α is supported in φ j (B(e, λ)), then by Definition 4.9, the norms are equivalent for −k m k, with equivalence constants independent of j. Thus, assertions (i) and (ii) follow straightforwardly from Theorem 5.14.
To get (iii), we only need to note that the operators in every degree. Indeed, by Proposition 2.14, the differential operator φ # j [χ j , d c ](φ −1 j ) # in H n has order 1 if h = n and order 0 if h = n. Since the kernel of T can be estimated by a kernel of type 2 if T acts on forms of degree h = n and of type 1 if it acts on forms of degree h = n, the assertion follows straightforwardly.
Summing up in j and taking into account that the sum is locally finite, we obtain Then statements (i)-(iii) follow straightforwardly from (i)-(iii) of Lemma 6.1.

Large scale geometry of contact sub-Riemannian manifolds
Theorems 1.2 and 1.5 are the key to proving that the validity of global Poincaré inequalities is equivalent to the vanishing of q, p cohomology, a large scale invariant of metric spaces. This equivalence will be established in [47]. By large scale invariant, we mean preserved, under uniform local assumptions, by quasi-isometries, i.e., maps f between metric spaces, which satisfy for suitable positive constants L and C.
Avoiding the general metric definition of q, p cohomology, let us give a construction valid for bounded geometry Riemannian manifolds with uniform vanishing of cohomology (the cohomology of an R -ball dies when restricted to a concentric R-ball, where the radius R depends only on the radius R). First, one defines the q, p cohomology of a simplicial complex: it is the quotient of the space of p simplicial cocycles by the image of q simplicial cochains by the coboundary operator. One shows that q, p cohomology is a quasi-isometry invariant of simplicial complexes with bounded geometry (i.e., bounded number of simplices through a vertex) and uniform vanishing of cohomology. Then one observes that every bounded geometry Riemannian manifold is quasi-isometric to such a simplicial complex.
Under similar boundedness and uniformity assumptions, one can show [47] that various locally acyclic complexes can be used to compute q, p cohomology. For contact sub-Riemannian manifolds, one can use either the exterior differential or Rumin's differential. As alluded to above, the building blocks are interior estimates and global smoothing, i.e., Theorems 1.2 and 1.5 and their Riemannian analogues. It follows that a global Poincaré inequality holds if and only if a global H-Poincaré inequality holds.
Using the Riemannian Hodge Laplacian, Müller et al. prove a Poincaré inequality Poincaré 2,q for the exterior differential on the Riemannian Heisenberg group [44,Lemma 11.2], under the assumption 1 2 − 1 q = 1 2n+1 . Therefore, their result combined with [47] provides an alternative proof of part of Corollary 1.4. We note that in degree h = n + 1, they miss the sharp exponent, given by our condition E(n + 1, 2, q, n).
The advantage of Rumin's Laplacian over its Riemannian sibling is its scale invariance. This allows us to apply the theory of singular integral operators to treat q, p cohomology for all p and to get the sharp exponent in degree h = n + 1. The drawback of the Rumin complex is that interior Poincaré inequalities become hard.

Three-dimensional Lie groups
There are four 3-dimensional Lie algebras that cannot be generated by a pair of vectors: the abelian Lie algebra R 3 , dil(2), the direct sum dil(1) ⊕ R, where dil(n) denotes the Lie algebra of the group of dilations and translations of R n , and the solvable unimodular Lie algebra sol. The Lie groups corresponding to other 3-dimensional Lie algebras admit left-invariant contact structures. All left-invariant sub-Riemannian metrics have bounded geometry, so Theorem 1.5 applies. When simply connected, they satisfy all uniform local assumptions required for the identification of the H-Poincaré p,q inequality with vanishing of q, p cohomology and its quasi-isometry invariance.
Here are examples.
The Heisenberg group H 1 is covered by Theorem 1.1. Note that the corresponding facts about q, p cohomology are new. M 2 := SL(2, R), the universal covering of SL(2, R), is quasi-isometric to PSL(2, R) × R. In degree 1, its p, p -cohomology vanishes for all p > 1; see [46]. Since PSL(2, R) acts isometrically and simply transitively on hyperbolic plane H 2 , it is quasi-isometric to H 2 . Since the p, p -cohomology of H 2 in degree 1 is Hausdorff and non-zero, the Künneth formula of [29] applies, and the p, p -cohomology in degree 2 of the product does not vanish because the p, p -cohomology in degree 1 of the line does not vanish. We conclude that, assuming 1 < p < ∞, the H-Poincaré p, p inequality holds in degree 1 and only in degree 1.

Other examples
Next, we describe a few non-simply-connected examples. Then the quasi-isometry invariance holds only in degree 1.
Let M 0 be the quotient of the Heisenberg group H 1 by the discrete subgroup generated by two elements, one of which belongs to the center of H 1 . Let us equip it with the quotient contact structure and sub-Riemannian metric.
is contained in a connected subgroup L of H 1 isomorphic to R 2 . This gives rise to a fibration M 0 → L \ H 1 , which is a line. The fibers of this map are tori with uniformly bounded diameters; therefore it is a quasi-isometry. The q, p cohomology of the line is well understood; it vanishes only when (q, p) = (∞, 1). Therefore, assuming 1 < p q < ∞, the H-Poincaré p,q inequality never holds for M 0 in degree 1.
Let M 1 denote the unit cotangent bundle of the Euclidean plane E 2 . It carries a tautological contact structure. The group G 1 of motions of the Euclidean plane, which is a semi-direct product of R 2 with S O(2), acts simply transitively on M, preserving the contact structure. Pick a G 1 -invariant sub-Riemannian metric on M. By invariance, the bounded geometry assumption is satisfied. The projection M → E 2 has uniformly bounded fibers; it is a quasi-isometry. Therefore M and E 2 have isomorphic exact q, p cohomologies in degree 1. The q, p cohomology of E 2 is well understood. It vanishes if and only if 1 p − 1 q 1 2 . We conclude that, assuming 1 < p q < ∞, the H-Poincaré p,q inequality holds for M 1 in degree 1 if and only if 1 p − 1 q 1 2 . Let us replace the Euclidean plane with the hyperbolic plane H 2 . The construction is identical up to the structure of the identity component G 2 of the isometry group of the hyperbolic plane: it is isomorphic to PSL(2, R). The obtained sub-Riemannian manifold M 2 is quasi-isometric to H 2 . The q, p cohomology of H 2 in degree 1 is well understood.
It vanishes only for p = 1. We conclude that the H-Poincaré p,q inequality never holds in degree 1 for M 2 if 1 < p q < ∞.

Further remarks
In each degree k, for every p, there is an exponent q = q(n, k) such that the L q -norm of Rumin k-forms is a conformal invariant (q(n, k) = 2n+2 k if k n, q(n, k) = 2n+2 k+1 if k n + 1). Therefore, in degree k, q(n,k−1),q(n,k) cohomology of 2n + 1-dimensional contact sub-Riemannian manifolds is a quasi-conformal invariant, and so is does the validity of an H-Poincaré q(n,k),q(n,k−1) inequality. We note that if k < 2n + 1, for the Heisenberg group H n , these cohomology groups vanish, whereas they need not vanish for other examples. For instance, if n = 1, q(n, 1) = 4 and q(n, 2) = 2, the 4,2 -cohomology in degree 2 of M 1 does not vanish. This shows that M 1 is not quasi-conformally equivalent to H 1 .
We see that Theorems 1.2 and 1.5 constitute useful tools for the geometric study of mappings between contact sub-Riemannian manifolds. Here are a few references about this emerging subject: [35] shows that two ways to take a quotient of a Heisenberg group by an isometry give rise to contact sub-Riemannian manifolds, which are not quasi-conformal. Moreover, [31] establishes the basic properties of quasi-regular maps, a study that has been continued in [12,19,37].