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 contact manifolds,” 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 d c 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.