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

TL;DR

This paper establishes well-posedness for transport and continuity equations in Heisenberg groups with contact vector fields, extending classical Euclidean results to a sub-Riemannian setting using adapted mollification techniques.

Contribution

It provides the first well-posedness results in a genuine sub-Riemannian context for contact vector fields with horizontal Sobolev regularity, adapting mollification strategies to Heisenberg groups.

Findings

First well-posedness in Heisenberg groups for contact vector fields

Extension of mollification techniques to sub-Riemannian geometry

Comparison with Euclidean BV case and alternative strategies

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 \`a la Bakry-\'Emery and an integral representation of the symmetrized derivative of $\mathbf b$.