Poincaré and sobolev inequalities for differential forms in heisenberg groups and contact manifolds

In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups, 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, which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of. 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.