Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups

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 $mc H^1$-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.