Abstract
AbstractIn this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is$${\mathscr {P}}_h$$Ph-rectifiable, for$$h\in {\mathbb {N}}$$h∈N, if it has positiveh-lower density and finiteh-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare$${\mathscr {P}}_h$$Ph-rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups, and we prove that it is strictly weaker than them. Second, we prove several structure properties of$${\mathscr {P}}_h$$Ph-rectifiable measures. Namely, we prove that the support of a$${\mathscr {P}}_h$$Ph-rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of$${\mathscr {P}}_h$$Ph-rectifiable measures with complemented tangents, we show that they are supported on the union of intrinsically Lipschitz and differentiable graphs. Such a covering property is used to prove the main result of this paper: we show that a$${\mathscr {P}}_h$$Ph-rectifiable measure has almost everywhere positive and finiteh-density whenever the tangents admit at least one complementary subgroup.
Funder
H2020 European Research Council
Simons Foundation
Publisher
Springer Science and Business Media LLC
Reference44 articles.
1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, pp. xviii+434 (2000). ISBN 0-19-850245-1
2. Ambrosio, L., Kirchheim, B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318(3), 527–555 (2000). (issn: 0025-5831)
3. Antonelli, G., Di Donato, D., Don, S., Le Donne, E.: Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups. Ann. l’Inst. Fourier. arXiv:2005.11390 (2020)
4. Antonelli, G., Merlo, A.: Intrinsically Lipschitz functions with normal target in Carnot groups. Ann. Fenn. Math. 46, 571–579 (2021)
5. Antonelli, G., Merlo, A.: On rectifiable measures in Carnot groups: Marstrand–Mattila rectifiability criterion. J. Funct. Anal. 283(1), 109495 (2022)
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献