# Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups

created by franchib on 20 Apr 2015

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Submitted Paper

Inserted: 20 apr 2015
Last Updated: 20 apr 2015

Year: 2015

Abstract:

The $L^1$-Sobolev inequality states that for compactly supported functions $u$ on the Euclidean $n$-space, the $L^{n/(n-1)}$-norm of a compactly supported function 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 $\mathcal H^1$-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.