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

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 $\mathbf b$.

💡 Research Summary

The paper addresses the well‑posedness of the transport and continuity equations on the Heisenberg groups (\mathbb{H}^n) for a class of vector fields that are contact and possess only horizontal Sobolev regularity. In the Euclidean setting, the theory of DiPerna–Lions (Sobolev vector fields) and Ambrosio (BV vector fields) provides a robust framework: one proves a renormalization property for weak solutions, which in turn yields existence, uniqueness, and the existence of a regular Lagrangian flow. However, these results rely heavily on the commutative structure of (\mathbb{R}^N) and cannot be transferred directly to sub‑Riemannian spaces where the horizontal distribution is non‑integrable and the Lie algebra is non‑abelian.

The authors consider a possibly time‑dependent vector field (b(t,\cdot)) on (\mathbb{H}^n) that satisfies the contact condition. Explicitly, there exists a generating function (\psi(t,\cdot)) such that \