Averages and the ℓ q,1 cohomology of Heisenberg groups

Abstract

Averages are invariants defined on the ℓ 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 ℓ 1 cohomology vanishes in these cases.R n ∂u ∂x n dx 1 · · · dx n = 0.A similar argument applies to other coordinates. Note that a n dx 1 ∧ · · · ∧ dx n = (−1) n−1 ω ∧ (dx 1 ∧ · · · ∧ dx n−1 ). 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 byThe 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 cohomology1.2. ℓ q,1 cohomology. It turns out that L 1,1 -cohomology has a topological content. By definition, the ℓ q,p cohomology of a bounded geometry Riemannian manifold is the ℓ 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 ℓ 1,1 -cohomology.We do not need define the ℓ q,p cohomology of simplicial complexes here, since, according to Theorem 3.3 of [6], every ℓ 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 ℓ p with p > 1, and especially ℓ 2 cohomology of Lie groups has been computed and used for large families of Lie groups, very little is known about ℓ 1 cohomology.

L1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

“…Remark 6.15. As in [8], Section 1.2, we stress that Poincaré inequality fails to hold in top degree (see also [34]).…”

“…This is the source of many complications. In particular, the classical Leibniz formula for the de Rham complex d(α ∧ β) = dα ∧ β ± α ∧ dβ is true in Rumin's complex only in special degrees, as shown in [13], Proposition A.1 and [34], Proposition 4.1. However, in general, the Leibniz formula fails to hold (see [13]-Proposition A.7).…”

Poincaré and Sobolev Inequalities for Differential Forms in Heisenberg Groups and Contact Manifolds

“…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].…”

The Rumin complex on nilpotent Lie groups

“…This problem can be avoided by considering a homotopically equivalent subcomplex (E • 0 , d c ) known as the Rumin complex, which better reads the homogeneity of the underlying Carnot group. We refer to [1,13,4,2,3,12] for the main results relating to hypoelliptic Laplacians on Carnot groups.…”