Poincaré and Sobolev inequalities for differential forms in Heisenberg groups and contact manifolds. (English) Zbl 1497.58001
Authors’ abstract: In this paper, we prove contact Poincare 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.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)
MSC:
| 58A10 | Differential forms in global analysis |
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
| 53D10 | Contact manifolds (general theory) |
| 26D15 | Inequalities for sums, series and integrals |
| 43A80 | Analysis on other specific Lie groups |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Heisenberg groups; differential forms; Sobolev-Poincaré inequalities; contact manifolds; homotopy formulaReferences:
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