Abstract
AbstractIn this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$, where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$, which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$. In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.
Highlights
We prove global H-Poincareand H-Sobolev inequalities and interior H-Poincareand 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 dc
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
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’
Summary
The Poincare 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 cu such that u − cu q C p,q,n du p. 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 φqCωp. 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 Poincare inequality
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