Averages and the $\ell^{q,1}$-cohomology of Heisenberg groups

Averages are invariants defined on the $\ell^1$ cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the $\ell^1$ cohomology vanishes in these cases.

Cet article est mis à disposition selon les termes de la licence C C -4.0 F .

From isoperimetry to averages of L 1 forms
The classical isoperimetric inequality implies that if u is a compactly supported function on R n , u n ≤ C du 1 , where n = n n−1 . Equivalently, every compactly supported closed 1-form ω admits a primitive u such that u n ≤ C ω 1 . More generally, if ω is a closed 1-form on R n which belongs to L 1 , does it have a primitive in L n ?
There is an obstruction. We observe that if each component a i of ω = n i=1 a i dx i is again in L 1 , the integral ∫ R n a i dx 1 . . . dx n is well defined and it is an obstruction for ω to be the differential of an L q function (for every finite q). Indeed, if ω = du, a n = ∂u ∂x n . For almost every (x 1 , . . . , x n−1 ), the function t → ∂u ∂x n (x 1 , . . . , x n−1 , t) belongs to L 1 and t → u(x 1 , . . . , x n−1 , t) belongs to L q . Since u(x 1 , . . . , x n−1 , t) tends to 0 along subsequences tending to +∞ or −∞, ∫ R ∂u ∂ x n (x 1 , . . . , x n−1 , t) dt = 0, hence More generally, if G is a Lie group of dimension n, there is a pairing, the average pairing, between closed L 1 k-forms ω and closed left-invariant (n − k)-forms β, defined by The integral vanishes if either ω = dφ where φ ∈ L 1 , or β = dα where α is left-invariant. Indeed, Stokes formula ∫ M dγ = 0 holds for every complete Riemannian manifold M and every L 1 form γ such that dγ ∈ L 1 . Hence the pairing descends to quotients, the L 1,1 cohomology L 1,1 H k (G) = closed L 1 k-forms/d(L 1 (k − 1)-forms with differential in L 1 ), and the Lie algebra cohomology H n−k (g) = closed, left-invariant (n − k)-forms/d(left-invariant (n − k − 1)-forms).

q,1 cohomology
It turns out that L 1,1 cohomology has a topological content. By definition, the q, p cohomology of a bounded geometry Riemannian manifold is the q, p cohomology of every bounded geometry simplicial complex quasiisometric to it. For instance, of a bounded geometry triangulation. Contractible Lie groups are examples of bounded geometry Riemannian manifolds for which L 1,1 cohomology is isomorphic to 1,1 cohomology.
We do not need define the q, p cohomology of simplicial complexes here, since, according to Theorem 3.3 of [7], every q, p cohomology class of a contractible Lie group can be represented by a form ω which belongs to L p as well as an arbitrary finite number of its derivatives. If the class vanishes, then there exists a primitive φ of ω which belongs to L q as well as an arbitrary finite number of its derivatives. This holds for all 1 ≤ p ≤ q ≤ ∞.
Although p with p > 1, and especially 2 cohomology of Lie groups has been computed and used for large families of Lie groups, very little is known about 1 cohomology.
1.3. From 1,1 to q,1 cohomology For instance, the averaging pairing is specific to 1 cohomology and it has never been studied yet. The first question we want to address is whether the averaging pairing provides information on q,1 cohomology for certain q > 1. Question 1.1. Given a Lie group G, for which exponents q and which degrees k is the averaging pairing q,1 H k (G) ⊗ H n−k (g) → R well-defined?
The question is whether there exists q > 1 such that the pairing vanishes on all L 1 forms which are differentials of L q forms. We just saw that for abelian groups R n , the pairing is well defined for k = 1 and all finite exponents q. Here is a more general result. Theorem 1.2. Let G be a Carnot group. In each degree 1 ≤ k ≤ n, there is an explicit exponent q(G, k) > 1 (see Definition 3.6) such that the averaging pairing is defined on We shall see that q(R n , k) = n = n n−1 in all degrees. For Heisenberg groups, q(H 2m+1 , k) = 2m+2 2m+1 if k m + 1, and q(H 2m+1 , m + 1) = 2m+2 2m .

Vanishing of the averaging pairing
The second question we want to address is whether the averaging pairing is trivial or not.
The pairing is always nonzero in top degree k = n. Indeed, there exist L 1 n-forms (even compactly supported ones) with nonvanishing integral. However, this seems not to be the case in lower degrees. Theorem 1.4. Let G be an abelian group or a Heisenberg group of dimension n. In each degree 1 ≤ k < n, the averaging pairing vanishes on q,1 H k (G) for q ∈ [1, q(G, k)].
In combination with results of [2], Theorem 1.4 implies a vanishing theorem for q,1 cohomology. In fact, [2] shows how in the Heisenberg case (and more trivially in the abelian case) it is possible to construct L q(G,k) primitives for d c -closed L 1 forms with vanishing averages, for all degrees k except top degree. It should be stressed that in proving this result, the vanishing of averages is a necessary extra assumption. Corollary 1.5. Let G be an abelian group or a Heisenberg group of dimension n. In each degree 0 ≤ k < n, q,1 H k (G) = 0 for q ≥ q(G, k). This is sharp. It is shown in [7] that q,1 H k (G) 0 if q < q(G, k). Also, in top degree, not only is q(G,n),1 H n (G) 0, but the kernel of the averaging map q(G,n),1 H n (G) → R = H 0 (g) * does not vanish. This is in contrast with the results of [1] in the case where p > 1, where it is proved that q(G,k), p H k (G) = 0 for all degrees k, including top degree. The results of [2] rely in an essential manner on analysis of the Laplacian on L 1 , inaugurated by J. Bourgain and H. Brezis, [4], adapted to homogeneous groups by S. Chanillo and J. van Schaftingen, [5].

Methods
The Euclidean space R n is n-parabolic, meaning that there exist smooth compactly supported functions ξ on R n taking value 1 on arbitrarily large balls, and whose gradient has an arbitrarily small L n norm. If ω is a closed L 1 form and β a constant coefficient form, and if ω = dψ, ψ ∈ L n , Stokes theorem gives which can be made arbitrarily small. This argument extends to Carnot groups of homogeneous dimension Q, which are Qparabolic. For this, one uses Rumin's complex, which has better homogeneity properties under Carnot dilations than de Rham's complex. When Rumin's complex is exactly homogeneous (e.g. for Heisenberg groups in all degrees, only for certain degrees in general), one gets a sharp exponent q(G, k). This leads to Theorem 1.2.
In Euclidean space, every constant coefficient form β has a primitive α with linear coefficients (for instance, dx 1 ∧ · · · ∧ dx k = d(x 1 dx 2 ∧ · · · ∧ dx k )). On the other hand, there exist cut-offs which decay like the inverse of the distance to the origin. Therefore which tends to 0. This argument extends to Heisenberg groups in all but one degree. To complete the proof of Theorem 1.4, one performs the symmetric integration by parts, integrating ω instead of β. For this, one produces primitives of ω on annuli, of linear growth.

Organization of the paper
In Section 2, the needed cut-offs are constructed. Theorem 1.2 is proven in Section 3. In order to integrate ω ∧ β by parts, one needs understand the behaviour of wedge products in Rumin's complex on Heisenberg groups, this is performed in Section 4. Section 5 exploits the linear growth primitives of left-invariant forms. In Section 6, controlled primitives of L 1 forms are designed, completing the proof of Theorem 1.4.

Annuli
We first construct cut-offs with an L ∞ control on derivatives. In a Carnot group G, we fix a subRiemannian metric and denote by B(R) the ball with center e and radius R. We fix an orthonormal basis of horizontal left-invariant vector fields W 1 , . . . , W n 1 . Given a smooth function u on G, and an integer m ∈ N, we denote by ∇ m u the collection of order . . , n 1 } m , and by |∇ m u| 2 the sum of their squares.

Lemma 2.2. Given f a (vector valued) function which is homogeneous of degree
Proof.
By applying a horizontal derivative to the left hand side of the equation, namely ∇ = W j with j ∈ {1, . . . , n 1 }, we get If we now apply ∇ to the right hand side, we get We have therefore proved that d f δ λ p = λ d−1 · d f p when restricted to horizontal derivatives, so we finally get our result

Parabolicity
Second, we construct cut-offs with a sharper L Q control on derivatives.
Let r be a smooth, positive function on G \ {e} that is homogeneous of degree 1 under dilations (one could think of a CC-distance from the origin, but smooth) and let us define the following function One should notice that χ(λR) = 0, χ(R) = 1, and that χ is smooth.
Using the smooth function χ introduced in (2.1), we can then define the cut-off function ξ as follows Lemma 2.4. The cut-off function ξ defined above has the following property: for every Proof. We compute otherwise.
tends to 0 as λ tends to infinity, provided m < Q. Let us stress that, in general, for values of m greater than or equal to Q, the final limit will not be zero. However, the values of m which we will be considering are very specific. This estimate will in fact be used in the proof of Proposition 3.3, and in that setting the degrees m that can arise will be all the possible degrees (or equivalently weights) of the differential d c on an arbitrary Rumin k-form φ of weight w. If we denote by M the maximal m that could arise in this situation, then one can show that M < Q.
Let us first assume that the maximal order M for the d c on k-forms (which is non trivial for 0 ≤ k < n) is greater or equal to Q. Then, given φ a Rumin k-form of weight Therefore M < Q, which means that indeed in all the cases that we will take into consideration, the L Q/m norm of ∇ m ξ will always go to zero as λ → ∞.
Remark 2.5. One says that a Riemannian manifold M is p-parabolic (see [8]) if for every compact set K, there exist smooth compactly supported functions on M taking value 1 on K whose gradient has an arbitrarily small L p norm. The definition obviously extends to subRiemannian manifolds. Lemma 2.4 implies that a Carnot group of homogeneous dimension Q is Q-parabolic. Let d c = j d c, j be the decomposition of d c into weights/orders. Let J (k, w) denote the set of weights/orders j such that d c, j on k-forms of weight w is nonzero, in other words,

The averaging map in general Carnot groups descends to cohomology
We will denote by J (k) the set of all the possible weights/orders, that is Let us define L χ(k) as follows and if J (k − 1,ŵ) = ∅ for someŵ, then we don't require anything on φŵ.

Lemma 3.2. Let G be a Carnot group. Fix a left-invariant subRiemannian metric making
be the decomposition of d c into weights (d c, j increases weights of Rumin forms by j, hence it has horizontal order j). Then the decomposition of d * c on a (k + 1)-form into weights/orders is In other words, the adjoint d * c, j of d c, j decreases weights by j and has horizontal order j. In fact, if we denote by J * (k + 1, w) the set of weights/orders j such that d * c, j on (k + 1)-forms of weight w is non-zero, that is then there is a clear relationship between the sets of indices J (k, w) and J * (k + 1, w), namely And from this relationship, we get directly the following identity: Moreover, since the formula d * c = (−1) n(k+1)+1 * d c * applies to any Rumin k-form, we also have the equality J * (n − k, Q − w) = J (k, w).

Proposition 3.3.
If ω, φ, β are Rumin forms with ω ∈ L 1 of degree k, β left-invariant of complementary degree n − k, φ ∈ L χ(k) and d c φ = ω, then Proof. Without loss of generality, one can assume that β has pure weight Q − w for some w ∈ W(k). Then its Hodge-star * β has pure weight w. Let ξ be a smooth cut-off. By definition of the L 2 -adjoint, we have Since φ ∈ L χ(k) , for any j ∈ J * (w − j) we have φ w−j ∈ L Q/Q−j , by definition of L χ(k) . Hence, applying Hölder's inequality, we obtain It is therefore sufficient to take ξ as the cut-off function introduced in Definition 2.3, so that by Lemma 2.4 we get that ∫ G ω ∧ β = 0. 3.1. Link with q,1 cohomology Let 1 ≤ p ≤ q ≤ ∞. According to Theorem 3.3 of [7], every q, p cohomology class of a Carnot group contains a form ω which belongs to L p as well as an arbitrary finite number of its derivatives. If the class vanishes, then there exists a primitive φ of ω which belongs to L q as well as an arbitrary finite number of its derivatives. There exists a homotopy between de Rham and Rumin's complex given by differential operators, therefore the same statement applies to Rumin's complex. In particular, Rumin forms can be used to compute q, p cohomology.
Let ω be a Rumin k-form which belongs to L 1 as well as a large number of its horizontal derivatives. Assume that ω represents the trivial cohomology class. Then there exists a Rumin (k − 1)-form φ which belongs to L q as well as its horizontal derivatives up to order Q, and such that d c φ = ω. By Sobolev's embedding theorem, φ belongs to L ∞ , hence to L q for all q ≥ q. This suggests the following notation. Proof of Theorem 1.2. Let G be a Carnot group. Let ω be a Rumin k-form on G, which belongs to L 1 as well as a large number of its derivatives. Assume that ω = d c φ with φ ∈ L q(G,k) . Then therefore φ ∈ L χ(k) . Proposition 3.3 implies that averages ∫ ω ∧ β vanish. This completes the proof of Theorem 1.2.  For these examples, as we saw before, one need not invoke [7] since J (k) has only one element in each degree. Example 3.9. Engel group E 4 . Then j(k) = 1 in degrees 1 and 4, j(k) = 2 in degrees 2 and 3. One concludes that the averaging map is well-defined in q,1 cohomology for q ≤ Q Q−1 in degrees 1 and 4, and for q ≤ Q Q−2 in degrees 2 and 3. Here, Q = 7.

Results for Heisenberg groups H 2m+1
In [2], it is proven that every closed L 1 k-form, k ≤ 2m, whose averages ∫ ω ∧ β vanish, is the differential of a form in L q , where q = q(k) = Q Q−1 unless when k = m + 1, where q(m + 1) = Q Q−2 . In other words, The goal of subsequent sections is to prove that the image of averaging map is 0 in all degrees k ≤ 2m. This will prove that L q(k),1 H k (G) = 0. Note that for k = 2m + 1, both facts fail: the averaging map is not zero (one can check with compactly supported forms) and it is not injective either (see [2]).

Wedge products between Rumin forms in Heisenberg groups
We shall rely on Stokes formula on Heisenberg groups H 2m+1 . We need a formula of the form d(φ ∧ β) = (d c φ) ∧ β ± φ ∧ d c β. This does not always hold in general for Carnot groups. In fact, the complex of Rumin forms E • 0 equals the Lie algebra cohomology H • (g), and therefore carries a natural cup product induced by the wedge product, but which in general differs from the wedge product. We shall see that in the case of Heisenberg groups, the difference does not show up in many degrees. This has already been investigated in [3].
Let us take into consideration the original construction of the Rumin complex in the (2m + 1)-dimensional Heisenberg group H 2m+1 as appears in [9].
Given Ω • the algebra of smooth differential forms, one can define the following two differential ideals: • I • := {α = γ 1 ∧ τ + γ 2 ∧ dτ}, the differential ideal generated by the contact form τ, and Remark 4.1. By construction, the ideal J • is in fact the annihilator of I • . In other words, given two arbitrary forms α ∈ J • and β ∈ I • , we have α ∧ β=0.
One can quickly check that the subspaces J h = J • ∩ Ω h are non-trivial for h ≥ m + 1, whereas the quotients Moreover, the usual exterior differential descends to the quotients Ω • /I • and restricts to the subspaces J • as first order differential operators: In [9] Rumin then defines a second order linear differential operator which connects the non-trivial quotients Ω • /I • with the non-trivial subspaces J • into a complex, that is d c • d c = 0,

Proposition 4.2. In H 2m+1 , the wedge product of Rumin forms is well-defined and satisfies the Leibniz rule
Proof. In order to study whether the wedge product between Rumin forms is well-defined, we will consider this differential operator d c in the following two cases: Let us first stress that in the first case, given α ∈ Ω h /I h , we have that It is important to notice that, however, if h+k = m, h > 0, k > 0, d c α∧ β+(−1) h α∧d c β involves only first derivatives of α and β, and thus cannot be equal to d c (α ∧ β). If h = 0 and k = m, d c (α ∧ β) involves second derivatives of α, and therefore cannot be expressed in terms of d c α.
On the other hand, in the second case, given β ∈ J h , the Rumin differential coincides with the usual exterior differential, Therefore, given α ∈ Ω h /I h and β ∈ J k with h < m and k > m, the wedge product α ∧ β is well-defined and belongs to J h+k , and the usual Leibniz rule also applies: If h = m and k ≥ m + 1, h + k ≥ 2m + 1, so the identity between differentials holds trivially.
To conclude, the wedge product of Rumin forms is well defined and satisfies the Proof. Let β ∈ E h 0 be a left-invariant form. Then d c β = 0, and β has weight w = h (if h ≤ m) or h + 1 (if h > m + 1). We know that the Rumin complex is locally exact, that is ∃ α ∈ E h−1 0 such that d c α = β. Let us consider the Taylor expansion of α at the origin in exponential coordinates, and let us group terms according to their homogeneity under dilations δ t :

Averages on
where we denote by α d the term with homogeneous degree d, i.e. δ * t α d = t d α d . Since d c commutes with the dilations δ t , the expansion of d c α is therefore The expansion of β is given instead by β = β, since it is a left-invariant form, hence homogeneous of degree w, so that β = d c α w .
Let us notice that α w has degree h − 1 and h m + 1, so it has weight w − 1. Since it is homogeneous of degree w under δ λ , its coefficients are homogeneous of degree 1, that is they are linear in horizontal coordinates, hence α w has linear growth, that is |α| ≤ C r.
which does not depend on the choice of primitive φ.

q,1 cohomology of bounded geometry Riemannian and subRiemannian manifolds
By definition, the q, p cohomology of a bounded geometry Riemannian manifold is the q, p cohomology of every bounded geometry simplicial complex quasiisometric to it. For instance, of a bounded geometry triangulation.
Combining results of [2] and Leray's acyclic covering theorem (in the form described in [6]), one gets that for q = n n−1 , the q,1 cohomology of a bounded geometry Riemannian n-manifold M is isomorphic to the quotient This applies in particular to Heisenberg groups H 2m+1 , and also to shells in Heisenberg groups, but with a loss on the width of the shell. 6.2. L 1 Poincaré inequality in shell B(λ) \ B(1) Lemma 6.1. There exist radii 0 < µ < 1 < λ < µ such that every d c -exact L 1 Rumin k-form ω on B(µ ) \ B(µ) admits a primitive φ on B(λ) \ B(1) such that In Euclidean space, the analogous statement can be proved as follows. Up to a biLipschitz change of coordinates, one replaces shells with products [0, 1] × S n−1 . On such a product, a differential form writes ω = a t + dt ∧ b t where a t and b t are differential forms on S n−1 . ω is closed if and only if each a t is closed and Given r ∈ [0, 1], define Note that each a r , and henceω, is an exact form on S n−1 . Since according to Subsection 6.1, there exists a formφ on S n−1 such that dφ =ω and Hence φ −φ is the required primitive. The Heisenberg group case reduces to the Euclidean case thanks to a smoothing homotopy constructed in [2]. In fact, since φ merely needs be estimated in L 1 norm (and not in the sharp L q norm), only the first, elementary, steps of [2] are required, resulting in the following result. Here, the W 1,1 norm refers to the L 1 norms of the first horizontal derivatives.
Proof. Pick a smooth function χ 1 with compact support in the large shell A. According to Lemma 6.2 of [2], there exists a left-invariant pseudodifferential operator K such that the identity holds on the space of Rumin forms K is the operator of convolution with a kernel k of type 1 (resp. 2 in degree n + 1). Using a cut-off, write k = k 1 + k 2 where k 1 has support in an -ball and k 2 is smooth. Since k 1 = O(r 1−Q ) or O(r 2−Q ) ∈ L 1 , the operator K 1 of convolution with k 1 is bounded on L 1 . Hence T = K 1 χ 1 is bounded on L 1 forms defined on A. Whereas S = d c K 2 χ 1 is bounded from L 1 to W s,1 for every integer s. If µ > λ + 2 and µ < 1 − 2 , the multiplication of ω by χ 1 has no effect on the restriction of d c K 1 ω or K 1 d c ω to the smaller shell, hence, in restriction to the smaller shell,

It follows that
and finally, in restriction to the smaller shell, Proof of Lemma 6.1. Using the exponential map, one can use simultaneously Heisenberg and Euclidean tools. Pick λ, µ, µ such that the medium Heisenberg shell B(µ −2 )\ B(µ+ 2 ) contains the Euclidean shell A eucl = B eucl (2) \ B eucl (1), which in turn contains the smaller Heisenberg shell B(λ) \ B(1). Apply Lemma 6.2 to a d c -closed L 1 form ω defined on the larger Heisenberg shell. Up to d c Tω, and up to restricting to the medium shell, one can replace ω with Sω which has its first horizontal derivatives in L 1 . Apply Rumin's homotopy Π E = 1 − dd −1 0 − d −1 0 d to get a usual d-closed differential form β = Π E Sω belonging to L 1 . Use the Euclidean version of Lemma 6.1 to get an L 1 primitive γ, dγ = β, on the Euclidean shell A eucl . Apply the order zero homotopy Π E 0 = 1 − d 0 d −1 0 − d −1 0 d 0 to get a Rumin form φ = Π E 0 γ. Its restriction to the smaller Heisenberg shell satisfies d c φ = ω and its L 1 norm is controlled by ω 1 .
Since δ * R commutes with the Rumin differential d c , we have Then, for ω R we have δ * R ω L 1 (B(λ)\B (1) where w is the weight of the k-form ω.
Only the first case is useful for our purpose.