Abstract
In 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.
A unique continuation theorem for exterior differential forms on Riemannian manifolds