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

Luigi Ambrosio, Gianluca Somma, Simone Verzellesi, Davide Vittone

TL;DR

This work addresses the well-posedness of the transport and continuity equations on the sub-Riemannian Heisenberg group $\mathbb{H}^n$ for time-dependent contact vector fields generated by $\psi\in W^{2,s}_{\mathcal{H}}(\mathbb{H}^n)$. It develops a geometry-adapted mollification framework, derives a precise integral representation for the commutator, and establishes a renormalization property that yields existence and uniqueness results, including Regular Lagrangian Flows, in this non-Euclidean setting. The results extend the DiPerna–Lions theory to sub-Riemannian geometry by exploiting cancellations from the contact structure, and they highlight the distinct behavior that arises from horizontal-vertical interactions, which cannot be captured by Euclidean BV theory or by Bakry–Émery interpolation alone. The paper also provides counterexamples and connections to existing frameworks, clarifying the limits of Euclidean and metric-measure approaches in the Heisenberg context. Overall, the work advances a robust well-posedness theory for transport-type PDEs in genuinely non-Euclidean spaces with structured velocity fields.

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 $\mathbf b$.

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

TL;DR

This work addresses the well-posedness of the transport and continuity equations on the sub-Riemannian Heisenberg group for time-dependent contact vector fields generated by . It develops a geometry-adapted mollification framework, derives a precise integral representation for the commutator, and establishes a renormalization property that yields existence and uniqueness results, including Regular Lagrangian Flows, in this non-Euclidean setting. The results extend the DiPerna–Lions theory to sub-Riemannian geometry by exploiting cancellations from the contact structure, and they highlight the distinct behavior that arises from horizontal-vertical interactions, which cannot be captured by Euclidean BV theory or by Bakry–Émery interpolation alone. The paper also provides counterexamples and connections to existing frameworks, clarifying the limits of Euclidean and metric-measure approaches in the Heisenberg context. Overall, the work advances a robust well-posedness theory for transport-type PDEs in genuinely non-Euclidean spaces with structured velocity fields.

Abstract

In this paper we obtain the well-posedness of the transport and continuity equations in the Heisenberg groups for a class of contact vector fields , under natural assumptions on the regularity of 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 geometry the mollification strategy of [18]. In the final part of the paper we illustrate why our result is not covered by the Euclidean 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 .
Paper Structure (19 sections, 22 theorems, 184 equations)

This paper contains 19 sections, 22 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main notation
  4. Heisenberg groups
  5. Group convolution
  6. Horizontal Sobolev spaces and contact vector fields
  7. Convergence of difference quotients
  8. Distributional solutions are renormalizable
  9. Transport equation and distributional solutions
  10. Renormalized solutions
  11. Regularization
  12. Proofs of \ref{['teo_distributional_implies_renormalized']} and \ref{['teo_uniqueness_null_solution']}
  13. Consequences of the renormalization property
  14. Transport equation with measurable initial conditions
  15. Regular Lagrangian Flows
  16. ...and 4 more sections

Key Result

Lemma 2.1

Define $\iota: {\mathbb{H}}^n\to{\mathbb{H}}^n$ by $\iota(q)=q^{-1}$. Then In particular, if $\varphi=\varphi\circ\iota$, then $Z_j\varphi=-Z_j^r\varphi\circ\iota$.

Theorems & Definitions (46)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 36 more