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

Baldi, Annalisa; Franchi, Bruno; Pansu, Pierre

doi:10.1016/j.aim.2020.107084

Cited by 12 publications

(22 citation statements)

References 54 publications

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“…Also, in top degree, not only is ℓ q(G,n),1 H n (G) = 0, but the kernel of the averaging map ℓ q(G,n),1 H n (G) → R = H 0 (g) * does not vanish. This is in contrast with the results of [1] concerning ℓ q,p H n (G) for p > 1, where nothing special happens in 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, [3], adapted to homogeneous groups by S. Chanillo and J. van Schaftingen, [4].…”

Section: Introductioncontrasting

confidence: 99%

“…In combination with results of [2], Theorem 1.2 implies a vanishing theorem for ℓ q,1 cohomology. Corollary 1.1.…”

Section: Introductionmentioning

confidence: 56%

“…Combining results of [2] and Leray's acyclic covering theorem (in the form described in [5]), one gets that for q = n n−1 , the ℓ q,1 -cohomology of a bounded geometry Riemannian n-manifold M is isomorphic to the quotient Similarly, if M is a bounded geometry contact subRiemannian manifold of dimension 2m + 1 (hence Hausdorff dimension Q = 2m + 2), for q = Q/(Q − 1) (respectively q = Q/(Q − 2) in degree m + 1), the ℓ q,1 cohomology of M is isomorphic to the quotient…”

Section: Averages On Heisenberg Group: Special Casementioning

confidence: 99%

“…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 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.…”

Section: The Averaging Map In General Carnot Groups Descends To Cohommentioning

confidence: 99%

“…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]).…”

Section: The Averaging Map In General Carnot Groups Descends To Cohommentioning

confidence: 99%

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Averages and the ℓ q,1 cohomology of Heisenberg groups

Pansu¹,

Tripaldi²

2020

Annales Mathématiques Blaise Pascal

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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.

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