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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.00804 |
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| _version_ | 1866910007526686720 |
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| author | Ambrosio, Luigi Somma, Gianluca Verzellesi, Simone Vittone, Davide |
| author_facet | Ambrosio, Luigi Somma, Gianluca Verzellesi, Simone Vittone, Davide |
| contents | 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 $\mathbf b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00804 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Renormalization of contact vector fields with horizontal Sobolev regularity in Heisenberg groups Ambrosio, Luigi Somma, Gianluca Verzellesi, Simone Vittone, Davide Analysis of PDEs 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 $\mathbf b$. |
| title | Renormalization of contact vector fields with horizontal Sobolev regularity in Heisenberg groups |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.00804 |