Preprint

Inserted: 31 jan 2026
Last Updated: 3 feb 2026

Year: 2026

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 $\mathbf b$.

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