Annalisa Baldi (Bologna University) – Hypoelliptic Laplacians and integral inequalities

In the last few years, in collaboration with B. Franchi (University of Bologna) and P. Pansu (University of Orsay), we have proved Poincaré and Sobolev inequalities in Heisenberg groups, for  Rumin differential forms. These inequalities can be seen as the analytical version of the well-known topological problem of whether a given closed form is exact. More precisely, one can ask whether a primitive can be upgraded to one that satisfies certain estimates; hence, geometric applications follow.

In the first part of the talk I would like to discuss the validity of these type of integral inequalities  in the Heisenberg groups, that descend from the existence of a fundamental solution of a suitable Hodge-type Laplacian defined by M. Rumin in this setting. The estimates that we obtain are sharp.  In the second part of the talk, I will discuss how we can be extended to general Carnot groups the same approach. The results presented in the second part include also  recent joint work with F. Tripaldi (University of Leeds) and A. Rosa (University of Bologna).