Antonelli Gioacchino, Merlo Andrea
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy.
Univerité Paris-Saclay, 307 Rue Michel Magat Bâtiment, 91400 Orsay, France.
J Geom Anal. 2022;32(9):239. doi: 10.1007/s12220-022-00971-7. Epub 2022 Jul 18.
In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is -rectifiable, for , if it has positive -lower density and finite -upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare -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 -rectifiable measures. Namely, we prove that the support of a -rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of -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 -rectifiable measure has almost everywhere positive and finite -density whenever the tangents admit at least one complementary subgroup.
在本文中，我们开始对卡诺群中一种新的可求长性概念进行详细研究：对于(s\in(0,Q))，我们称一个拉东测度是(s -)可求长的，如果它几乎处处具有正的(s -)下密度和有限的(s -)上密度，并且在几乎每一点处，它至多相差倍数具有唯一的切测度。首先，我们在卡诺群的背景下，将(s -)可求长性与文献中先前已知的其他可求长性概念进行比较，并证明它严格弱于它们。其次，我们证明了(s -)可求长测度的几个结构性质。具体而言，我们证明了(s -)可求长测度的支撑集几乎处处被满足锥状性质的集合所覆盖，并且在具有互补切空间的(s -)可求长测度的特殊情况下，我们表明它们支撑在内在利普希茨和可微图的并集上。这样一种覆盖性质被用于证明本文的主要结果：我们表明，当切空间至少有一个互补子群时，(s -)可求长测度几乎处处具有正的且有限的(s -)密度。
