arXiv subjects

Gianluca Somma

Publications and source records attributed to Gianluca Somma.

3 recordsLinked to original sources

arXiv2026

Transport and flow for horizontal Sobolev contact velocities in Carnot groups

We establish new well-posedness results for transport and flow equations driven by contact vector fields on Carnot groups. The velocity fields are assumed to have horizontal Sobolev regularity, namely Sobolev regularity only along the horizontal directions determined by the stratified geometry of the group. In the broader sub-Riemannian setting, results of this type were previously known only for Heisenberg groups. Our proof relies on the theory of renormalized solutions as originally introduced by DiPerna and Lions in the Euclidean setting.

math.AP↗

arXiv2026

A note on the diameter of small sub-Riemannian balls

We observe that the diameter of small (in a locally uniform sense) balls in $C^{1,1}$ sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to $C^0$, the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.

math.OC↗

arXiv2026

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 $\mathbf b$.

math.AP↗