We prove a rank-one theoremà la G. Alberti for the derivatives of vectorvalued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups H n for n ≥ 2. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph.2010 Mathematics Subject Classification. 49Q15, 28A75, 49Q20.

We study properties of functions with bounded variation in Carnot-Carathéodory spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative. IntroductionA lot of effort was devoted in the last decades to the development of Analysis and Geometry in general metric spaces and, in particular, to the study of functions with bounded variation (BV ). Carnot-Carathéodory (CC) spaces are among the most fruitful settings where BV functions have been introduced ([10, 20]), see also [8,12,19,22,23,24] and the more recent [3,5,6,9,11,15,31,35,44]. The aim of this paper is to give some contributions to this research lines by establishing "fine" properties of BV functions in CC spaces. A non-trivial part of our work consists in fixing the appropriate language in a consistent and robust manner.A CC space is the space R n endowed with the Carnot-Carathéodory distance d (see (1)) arising from a fixed family X = (X 1 , . . . , X m ) of smooth, linearly independent vector fields (called horizontal) in R n satisfying the Hörmander condition, see (2). As customary in the literature, we always assume that metric balls are bounded with respect to the Euclidean topology. Moreover, we work in equiregular CC spaces, where a homogeneous dimension Q, usually larger than the topological dimension n, can be defined; recall that any CC space can be lifted to an equiregular one, see e.g. [42].The space BV X of function with bounded X-variation consists of those functions u whose derivatives X 1 u, . . . , X m u in the sense of distributions are represented by a vector-valued measure D X u with finite total variation |D X u|. These functions have been extensively studied in the literature and important properties have been proved, like coarea formulae, approximation theorems, Poincaré inequalities.We now describe some of the results we prove in this paper. The first one, Theorem 1.1 below, concerns the almost everywhere approximate X-differentiability (see Section 2.3) of BV X functions; its classical counterpart is very well-known, see e.g. [2, Theorem 3.83]. As customary, we denote by D a X u and D s X u, respectively, the absolutely continuous and singular part of D X u with respect to the Lebesgue measure L n . (Italy) project "Campi vettoriali, superfici e perimetri in geometrie singolari". The second named author wishes to ackowledge the support and hospitality of FBK-CIRM (Trento), where part of this paper was written.

In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets Γ that, similarly to intrinsic Lipschitz graphs, have a cone property:there exists an open dilation-invariant subset C whose translations by elements in Γ don't intersect Γ. However, a priori the cone C may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second type.

We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $\Gamma$-liminf of the rescaled energy in terms of the horizontal perimeter.

We prove a compactness result for bounded sequences (u j ) j of functions with bounded variation in metric spaces (X, d j ) where the space X is fixed but the metric may vary with j. We also provide an application to Carnot-Carathéodory spaces.2010 Mathematics Subject Classification. 46E35, 26B30, 26D10, 53C17.