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POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 3 / May 2022
- Published online by Cambridge University Press:
- 29 June 2020, pp. 869-920
- Print publication:
- May 2022
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- Article
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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.
