SOBOLEV-POINCARE INEQUALITIES FOR DIFFERENTIAL FORMS AND CURRENTS IN R-n

In this Note we collect some results in R(n )about (p, q) Poincare and Sobolev inequalities, with 1 <= p < n, for differential forms obtained in a joint research with Franchi and Pansu. In particular, we focus to the the case p = 1 . From the geometric point of view, Poincare and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. As an application of the results obtained in the case p = 1 we obtain a Poincare and Sobolev inequalities for Euclidean currents