Francesca Tripaldi University of Leeds

Possible div-curl estimates on the 5-dimensional Cartan group

Abstract:

On arbitrary Carnot groups, the only hypoelliptic Hodge-Laplacians on forms that have been introduced are 0-order pseudodifferential operators constructed using the Rumin complex. However, to address questions where one needs sharp estimates, this 0-order operator is not suitable. Indeed, this is a rather difficult problem to tackle in full generality, the main issue being that the Rumin exterior differential is not homogeneous on arbitrary Carnot groups. In this talk, I will focus on the specific example of the free Carnot group of step 3 with 2 generators, where it is possible to introduce different hypoelliptic Hodge-Laplacians on forms. Such Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups.