Rank-one theorem and subgraphs of BV functions in Carnot groups

Abstract

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.

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

“…Remark 2.27. The proof of Proposition 2.26 implicitly shows that the validity of (20) does not depend on the choice of the function f used in (15) to define B ± ν (p, r). The proof of the following result is standard and we postpone it to the Appendix B.…”

“…For the converse implication one has to prove that, if (ii) holds and f is a C 1 X real function on a neighborhood of p such that f (p) = 0 and Xf (p)/|Xf (p)| = ν, then (20) holds with B ± ν (p, r) defined (see (15)) in terms of f . By Theorem 2.2 and a change of variables, proving (20) amounts to proving that…”

“…">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.…”

Area of intrinsic graphs and coarea formula in Carnot groups

“…[2,4,7,8,17]. A relevant tool in the proof of Theorem 1.1 is a differentiation theorem for measures (Proposition 2.2) which is based on the so-called Federer density (10): the importance of this notion was pointed out only recently by V. Magnani, see [39][40][41] and [20]. Observe that the validity of a (currently unavailable) Rademacher-type Theorem for intrinsic Lipschitz graphs would likely allow to extend Theorem 1.1 to the case of intrinsic Lipschitz φ.…”

“…Left translations and dilations of W → V intrinsic graphs are again W → V intrinsic graphs, see[4, Proposition 3.6].The proof of the following lemma is inspired by[10, Theorem A.5]. Similar statements are contained in[17, Theorem 3.27] and[38, Theorem 1.4].Lemma 2.10 (Implicit Function Theorem) Let 0 ⊂ G be open, g ∈ C 1 H ( 0 ; G ) and let o ∈ G be a split-regular point of g. Let G = W • V be a splitting of G such that ker(D H g(o)) is a (necessarily entire) 4 intrinsic graph W → V. Then there are neighborhoods A of π W (o) in W, B of g(o) in G and ⊂ 0 of o, and a map ϕ : A × B → V such that the map (a, b) → aϕ(a, b) is a homeomorphism A × B → and g(aϕ(a, b)) = b.…”