$L^1$-Poincar\'e inequalities for differential forms on Euclidean spaces and Heisenberg groups

Inégalités de Poincaré $L^1$ pour les formes différentielles sur les espaces euclidiens et les groupes d'Heisenberg

Annalisa Baldi (1) , Bruno Franchi (1) , Pierre Pansu (2)

1 Dipartimento di Matematica [Bologna]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

In this paper, we prove interior Poincaré and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-van Schaftingen in Heisenberg groups.

Domaines: Géométrie différentielle [math.DG]

Annalisa Baldi, Bruno Franchi, Pierre Pansu. $L^1$-Poincar\'e inequalities for differential forms on Euclidean spaces and Heisenberg groups. 2019. ⟨hal-02015047⟩