Besicovitch Covering Property on graded groups and applications to measure differentiation
Besicovitch Covering Property on graded groups and applications to measure differentiation
复制标题
分级组的贝西科维奇覆盖特性及测量差异化的应用
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Séverine Rigot
中科院分区：
文献类型：
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作者：
Enrico Le Donne;Séverine Rigot
我们给出了齐次群允许齐次距离满足Besicovitch覆盖性质（BCP）的完整答案。
保持。特别是，我们证明了一个分层组承认齐次距离BCP举行当且仅当该组有步骤1或2。这些结果是作为一个更普遍的研究均匀准距离分次群的后果。即证明了正分次群有满足BCP的连续齐次拟距离当且仅当它的李代数的相应正分次的任意两个不同层可交换。BCP的有效性有几个后果。它与测度差异理论的联系是本文的主要动机之一。作为我们的结果的结果，我们得到，例如，分层群可以配备一些齐次距离，使微分定理对每个局部有限的Borel测度成立当且仅当该组有步骤1或2。在本文中开发的技术也使我们能够证明，分黎曼距离分层组的步骤2或更高的从来没有满足BCP。使用爆破技术，这表明意味着在一个子黎曼流形上的微分定理不成立的一些局部有限的博雷尔措施。
We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP)
holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.