In the free, step-2, rank-4 sub-Riemannian Carnot group, we give a clean expression for length-extremals, we provide an explicit equation for conjugate points, we relate it with the conjectured cut-locus of the origin. Finally, we give some upper estimates for the cut-time of extremals.

We present an overview of the scientific activity of Ermanno Lanconelli, to whom this volume is dedicated on the occasion of his birthday.

We study the subRiemannian cut time and cut locus of a given point in a class of step-2 Carnot groups of Reiter-Heisenberg type. Following the Hamiltonian point of view, we write and analyze extremal curves, getting the cut time of any of them, and a precise description of the set of cut points.

In this paper, we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-n Carnot algebra is isomorphic to the exterior algebra of ℝ^n . Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.

A subset of a Carnot group is said to be precisely monotone if the restriction of its characteristic function to each integral curve of every left-invariant horizontal vector field is monotone. Equivalently, a precisely monotone set is a h-convex set with h-convex complement. Such sets have been introduced and classified in the Heisenberg setting by Cheeger and Kleiner in the 2010's. In the present paper, we study precisely monotone sets in the wider setting of step-2 Carnot groups, equivalently step-2 Carnot algebras. In addition to general properties, we prove a classification in terms of sublevel sets of h-affine functions in step-2 rank-3 Carnot algebras that can be seen as a generalization of the one obtained by Cheeger and Kleiner in the Heisenberg setting. There is however a significant difference here as it is known that, unlike the Heisenberg setting, there are sublevel sets of h-affine functions on the free step-2 rank-3 Carnot algebra that are not half-spaces.

In [Formula: see text] we consider the vector fields [Formula: see text] where [Formula: see text]. Let [Formula: see text] be the (closed) upper half-space and let [Formula: see text] be a function such that [Formula: see text] for some [Formula: see text]. In this paper, we prove that the restriction of [Formula: see text] to the plane [Formula: see text] belongs to a suitable Besov space that is defined using the Carnot–Carathéodory metric associated with [Formula: see text] and [Formula: see text] and the related perimeter measure.

We analyze some properties of a class ofmultiexponential mapsappearing naturally in the geometric analysis of Carnot groups. We will see that such maps can be useful in at least two interesting problems: first, in relation to the analysis of some regularity properties of horizontally convex sets. Then, we will show that our multiexponential maps can be used to prove thePansu differentiabilityof the subRiemannian distance from a fixed point.

In the setting of step two Carnot groups, we show a "cone property" for horizontally convex sets. Namely we prove that, given a horizontally convex set $C$, a pair of points $P\in \partial C$ and $Q\in $ int $C$, both belonging to a horizontal line $\ell$, then an open truncated subRiemannian cone around $\ell$ and with vertex at $P$ is contained in $C$. We apply our result to the problem of classification of horizontally monotone sets in Carnot groups. We are able to show that monotone sets in the direct product $\mathbb{H} \times\mathbb{R}$ of the Heisenberg group with the real line have hyperplanes as boundaries.

We discuss some estimates of subelliptic type related with vector fields satisfying the H\ormander condition. Our approach makes use of a class of approximate exponentials maps. Such kind of estimates arises naturally in the study of regularity theory of weak solutions of degenerate elliptic equations.

Given the pair of vector fields X = ∂x + |z|2my∂t and Y = ∂y −|z|2mx∂t,where (x,y,t) = , we give a condition on a bounded domain which ensures that Ω is an (ε,δ)-domain for the Carnot-Carathéodory metric. We also analyze the Ahlfors regularity of the natural surface measure induced on ∂Ω by the vector fields.

We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators, giving a new, independent proof of a result by Myasnichenko (J Dyn Control Syst 8(4):573-597, 2002). We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Furthermore, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.

We prove a quantitative openness theorem for C^1 submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carathéodory spaces, and we improve a classical completeness result by Palais.

We show by explicit estimates that the SubRiemannian distance in a Carnot group of step two is locally semiconcave away from the diagonal if and only if the group does not contain abnormal minimizing curves. Moreover, we prove that local semiconcavity fails to hold in the step-3 Engel group, even in the weaker “horizontal” sense.

In this survey we consider a general Hörmander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formula on the level set of the fundamental solution, which allow to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results: estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem.

Consider a family \(\mathcal{H}:= \{X_{j} =: f_{j}\cdot\nabla: j=1,\ldots , m\}\) of C 1 vector fields in ℝ n and let s∈ℕ. We assume that for all p∈{1,…,s} and j 1,…,j p ∈{1,…,m} the horizontal derivatives \(X_{j_{1}}X_{j_{2}}\cdots X_{j_{p-1}}f_{j_{p}}\) exist and are Lipschitz continuous with respect to the control distance defined by \(\mathcal{H}\). Then we show that different notions of commutator agree. This involves an accurate analysis of some algebraic identities involving nested commutators which seem to have an independent interest.

We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.

We consider a family of $C^1$ vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincaré inequality.

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details

We prove a ball-box theorem for nonsmooth Hormander vector fields of step s.