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.

中文翻译：

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海森堡群和接触歧管中微分形式的庞加莱和索博列夫不等式

在本文中，我们证明了 Heisenberg 群中的接触 Poincaré 和 Sobolev 不等式 $\mathbb{H}^{n}$ ，其中“接触”一词的意思是强调德拉姆的外部差异被所谓的鲁敏复合体的外部差异所取代 $(E_{0}^{\bullet },d_{c})$ , 它恢复了与李代数的分层相关的群膨胀下的尺度不变性 $\mathbb{H}^{n}$ . 此外，我们在有界几何的亚黎曼接触流形上构造了微分形式的平滑算子，这些算子对上同调的作用微乎其微。例如，这允许我们用更规则的微分形式替换封闭形式，直到添加受控精确形式。