Fine properties of functions with bounded variation in Carnot-Carathéodory spaces

Abstract

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.

On rectifiable measures in Carnot groups: representation

“…where the last identity comes from the injectivity of P V when restricted to , since α ≤ ε 1 (V, L), see Proposition 2. 13. Finally, Proposition 2.9 implies…”

“…Let us remark that the previous results in Proposition 1.4 and Corollary 1.5 follow from the rectifiability criterion in Proposition 3.9. It is worth pointing out that, given a Lipschitz function f : B ⊆ G → R, for every y ∈ f (B), the set { f ≤ y} is of locally finite perimeter in G, see, e.g., [13,Theorem 2.40]. Hence, taking into account Corollary 1.5, we deduce the following consequence: S 1 -almost all the sublevel sets of real-valued Lipschitz functions defined on Borel subsets of Carnot groups are examples of sets of locally finite perimeter whose boundary is C 1 H -rectifiable-namely De Giorgi's rectifiability Theorem holds for such sets.…”

“…By the very choice of σ this implies that the latter compact sets are C V i, (min{α, 1/ })-sets, and since 1/ < ε 1 (V i, , L i, ), we also conclude that they are graphs according to the splitting G = V i, • L i, , see Proposition 2. 13.…”

Local minimizers and gamma-convergence for nonlocal perimeters in Carnot groups

“…Motivated by these results, we introduce the following notation, see [21] that will be used in Theorem 4.9 and Remark 4.8. Also recall Definitions 2.8 and 2.5.…”

A compactness result for BV functions in metric spaces

“…Our interest in Theorem 3.6, in turn, was originally motivated by the study of fine properties of BV X functions in CC spaces and, in particular, of their local properties. Here, one often needs to perform a blow-up procedure around a fixed point p: it is well-known that this produces a sequence of CC metric spaces (R n , X j ) that converges to (a quotient of) a Carnot group structure G. In this blow-up, the original BV X function u 0 gives rise to a sequence (u j ) j of functions in BV X j which, up to a subsequence, will converge in L 1 loc to a BV G,loc function u in G. The function u (typically: a linear map, or a jump map taking two different values on complementary halfspaces of G) will then provide some information on u 0 around p. We refer to [3] for more details.…”