Renormalization of contact vector fields with horizontal Sobolev regularity in Heisenberg groups

Abstract

In this paper we obtain the well-posedness of the transport and continuity equations in the Heisenberg groups ${\mathbb{H}}^n$ for a class of contact vector fields $\mathbf{b}$, under natural assumptions on the regularity of $\mathbf{b}$ not covered by the, now classical, Euclidean theory [18]. It is the first example of well-posedness in a genuine sub-Riemannian setting, that we obtain adapting to the ${\mathbb{H}}^n$ geometry the mollification strategy of [18]. In the final part of the paper we illustrate why our result is not covered by the Euclidean $BV$ case solved by the first author in [1], and we compare it with the strategy of [7], based on the representation of the commutator by interpolation à la Bakry-Émery and an integral representation of the symmetrized derivative of $\mathbf{b}$.