L1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

Abstract

In this paper, we prove interior Poincar\ue9 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 L1 norm. Unlike for Lp, 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