$L^1$-Poincar\'e and Sobolev inequalities for differential forms in Euclidean spaces

TL;DR

This paper establishes $L^1$-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces, extending classical results by replacing singular integral estimates with Bourgain-Brezis type inequalities.

Contribution

It introduces new $L^1$-based inequalities for differential forms, broadening the scope of classical Sobolev and Poincaré inequalities in Euclidean spaces.

Findings

Proves $L^1$-Poincaré inequalities for differential forms.

Establishes Sobolev inequalities in the $L^1$ setting.

Replaces singular integral estimates with Bourgain-Brezis inequalities.

Abstract

In this paper, we prove Poincar\'e and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.