2019•arXiv (Cornell University)Open access

$L^1$-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

Annalisa Baldi, Bruno Franchi, Pierre Pansu

Abstract

In this paper, we prove interior Poincar{é} 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.

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Key concepts: Mathematics, Euclidean geometry, Heisenberg group, Pure mathematics, Poincaré conjecture, Euclidean space, Limiting, Differential (mechanical device)