ORLICZ SPACES AND ENDPOINT SOBOLEV-POINCAR ́EINEQUALITIES FOR DIFFERENTIAL FORMS INHEISENBERG GROUPS

In this paper we prove Poincar ́e and Sobolev inequalities for differ-ential forms in the Rumin’s contact complex on Heisenberg groups. Inparticular, we deal with endpoint values of the exponents, obtaining fi-nally estimates akin to exponential Trudinger inequalities for scalar func-tion. These results complete previous results obtained by the authors awayfrom the exponential case. From the geometric point of view, Poincar ́eand Sobolev inequalities for differential forms provide a quantitative for-mulation of the vanishing of the cohomology. They have also applicationsto regularity issues for partial differential equations.