Poincaré inequalities for differential forms on Heisenberg groups

Every closed differential form \omega on a Euclidean ball has a primitive whose L^q norm is bounded by the L^p norm of \omega (for suitable exponents p and q). We prove an analogous result for Rumin's exterior differential on Heisenberg balls. This is used to prove vanishing of \ell^{q,p}-cohomology of Heisenberg groups. Extension to other Carnot groups will be discussed.