Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates

On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered in (M. Rumin, “Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups,” C. R. Acad. Sci., Paris Sér. I Math., vol. 329, no. 11, pp. 985–990, 1999, M. Rumin, “Sub-Riemannian limit of the differential form spectrum of contactmanifolds,” Geom. Funct. Anal., vol. 10, no. 2, pp. 407–452, 2000), where the author introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge- Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential dc is not homogeneous on arbitrary Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians.We compare how these three possible 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.