卡诺群中的可校正测度:密度的存在性。

IF 1.2 2区 数学 Q1 MATHEMATICS Journal of Geometric Analysis Pub Date : 2022-01-01 Epub Date: 2022-07-18 DOI:10.1007/s12220-022-00971-7
Gioacchino Antonelli, Andrea Merlo
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引用次数: 9

摘要

本文详细研究了卡诺群中可纠偏性的一个新概念:对于h∈N,如果Radon测度几乎处处具有正的h-下密度和有限的h-上密度,并且在几乎每一点上,它都有一个唯一的可纠偏测度。首先,我们将h -可纠偏性与文献中已知的卡诺群背景下的其他可纠偏性概念进行了比较,证明了h -可纠偏性严格弱于它们。其次,我们证明了ph可整流措施的几个结构性质。也就是说,我们证明了h -可整流测度的支持几乎处处被满足锥状性质的集合所覆盖,并且在具有互补切线的h -可整流测度的特殊情况下,我们证明了它们在本质Lipschitz图与可微图的并集上是支持的。利用这一覆盖性质证明了本文的主要结果:我们证明了当切线至少有一个互补子群时,h可整流测度几乎处处具有正的有限h密度。
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On Rectifiable Measures in Carnot Groups: Existence of Density.

In this paper, we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is P h -rectifiable, for h N , if it has positive h-lower density and finite h-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples. First, we compare P h -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 P h -rectifiable measures. Namely, we prove that the support of a P h -rectifiable measure is almost everywhere covered by sets satisfying a cone-like property, and in the particular case of P h -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 P h -rectifiable measure has almost everywhere positive and finite h-density whenever the tangents admit at least one complementary subgroup.

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来源期刊
CiteScore
2.00
自引率
9.10%
发文量
290
审稿时长
3 months
期刊介绍: JGA publishes both research and high-level expository papers in geometric analysis and its applications. There are no restrictions on page length.
期刊最新文献
Multicomplexes on Carnot Groups and Their Associated Spectral Sequence. The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation. Worm Domains are not Gromov Hyperbolic. On the Normal Stability of Triharmonic Hypersurfaces in Space Forms. Horizontally Affine Functions on Step-2 Carnot Algebras.
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