A1 Journal article (refereed)
Besicovitch Covering Property on graded groups and applications to measure differentiation (2019)

Le Donne, E., & Rigot, S. (2019). Besicovitch Covering Property on graded groups and applications to measure differentiation. Journal für die Reine und Angewandte Mathematik, 2019(750), 241-297. https://doi.org/10.1515/crelle-2016-0051

JYU authors or editors

Le Donne, Enrico

Publication details

All authors or editors: Le Donne, Enrico; Rigot, Séverine

Journal or series: Journal für die Reine und Angewandte Mathematik

ISSN: 0075-4102

eISSN: 1435-5345

Publication year: 2019

Volume: 2019

Issue number: 750

Pages range: 241-297

Publisher: de Gruyter

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1515/crelle-2016-0051

Publication open access: Not open

Publication channel open access:

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1512.04936

Abstract

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.

Keywords: measure theory; harmonic analysis (mathematics)

Free keywords: covering theorems; Besicovitch Covering Property; measure differentiation

Fields of science: