Gagliardo-Nirenberg Inequalities for Differential Forms in Heisenberg Groups

TL;DR

This paper extends Gagliardo-Nirenberg inequalities for differential forms from Euclidean spaces to Heisenberg groups, broadening the scope of these inequalities in geometric analysis.

Contribution

It generalizes existing inequalities for differential forms to the setting of Heisenberg groups using an appropriate complex of differential forms.

Findings

Extended inequalities to Heisenberg groups.

Established control of L^{n/(n-1)}-norm by exterior derivatives.

Connected inequalities with Hardy norms in special cases.

Abstract

The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly supported differential h-form is controlled by the L 1-norm of its exterior differential du and its exterior codifferential $\delta$u (in special cases the L 1-norm must be replaced by the H 1-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.