  In the last few years the authors proved Poincar\'e and Sobolev type
inequalities in Heisenberg groups $\mathbb{H}^n$ for differential forms in the
Rumin's complex. The need to substitute the usual de Rham complex of
differential forms for Euclidean spaces with the Rumin's complex is due to the
different stratification of the Lie algebra of Heisenberg groups. The crucial
feature of Rumin's complex is that $d_c$ is a differential operator of order 1
or 2 according to the degree of the form. Roughly speaking, Poincar\'e and
Sobolev type inequalities are quantitative formulations of the well known
topological problem whether a closed form is exact. More precisely, for
suitable $p$ and $q$, we mean that every exact differential form $\omega$ in
$L^p$ admits a primitive $\phi$ in $L^q$ such that$\|\phi\|_{L^{q}}\leq C\
\|\omega\|_{L^{p}}$. The cases of the norm $L^p$, $p\ge 1$ and $q<\infty$ have
been already studied in a series of papers by the authors. In the present paper
we deal with the limiting case where $q=\infty$: it is remarkable that, unlike
in the scalar case, when the degree of the forms $\omega$ is at least $2$, we
can take $q=\infty$ in the left-hand side of the inequality. The corresponding
inequality in the Euclidean setting $\mathbb{R}^N$ ($p=N$ and $q=\infty$) was
proven by Bourgain and Brezis.
