Cohomology of annuli, duality and $L^\infty$-differential forms on Heisenberg groups

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.