Continuous primitives for higher degree differential forms in Euclidean spaces, Heisenberg groups and applications

Annalisa Baldi (1) , Bruno Franchi (1) , Pierre Pansu (2, 3)

1 Dipartimento di Matematica [Bologna]
2 DATASHAPE - Understanding the Shape of Data
3 LMO - Laboratoire de Mathématiques d'Orsay

It is shown that higher degree exact differential forms on compact Riemannian $n$-manifolds possess continuous primitives whose uniform norm is controlled by their $L^n$ norm. A contact sub-Riemannian analogue is proven, with differential forms replaced with Rumin differential forms.

Il est d\'emontr\'e que les formes diff\'erentielles exactes de degr\'e sup\'erieur sur des vari\'et\'es riemanniennes compactes de dimension $n$ poss\`edent des primitives continues dont la norme uniforme est contr\^ol\'ee par leur norme $L^n$. Un analogue dans des vari\'et\'es sous-riemanniennes de contact est prouv\'e, les formes diff\'erentielles \'etant remplac\'ees par des formes diff\'erentielles de Rumin.

Mots clés: Heisenberg groups; differential forms; Poincar\'e inequalities; Beppo Levi spaces

Domaines: Equations aux dérivées partielles [math.AP]

Annalisa Baldi, Bruno Franchi, Pierre Pansu. Continuous primitives for higher degree differential forms in Euclidean spaces, Heisenberg groups and applications. 2024. ⟨hal-04518692⟩