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Remarks about the Besicovitch Covering Property in Carnot groups of step 3 and higher


Authors: Enrico Le Donne and Séverine Rigot
Journal: Proc. Amer. Math. Soc. 144 (2016), 2003-2013
MSC (2010): Primary 28C15, 49Q15, 43A80
DOI: https://doi.org/10.1090/proc/12840
Published electronically: September 15, 2015
MathSciNet review: 3460162
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Abstract: We prove that the Besicovitch Covering Property (BCP) does not hold for some classes of homogeneous quasi-distances on Carnot groups of step 3 and higher. As a special case we get that, in Carnot groups of step 3 and higher, BCP is not satisfied for those homogeneous distances whose unit ball centered at the origin coincides with a Euclidean ball centered at the origin. This result comes in contrast with the case of the Heisenberg groups where such distances satisfy BCP.


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Additional Information

Enrico Le Donne
Affiliation: Department of Mathematics and Statistics, P.O. Box 35, FI-40014, University of Jyväskylä, Finland
Email: ledonne@msri.org

Séverine Rigot
Affiliation: Laboratoire de Mathématiques J.A. Dieudonné UMR CNRS 7351, Université Nice Sophia Antipolis, 06108 Nice Cedex 02, France
Email: rigot@unice.fr

DOI: https://doi.org/10.1090/proc/12840
Keywords: Covering theorems, Carnot groups, homogeneous quasi-distances
Received by editor(s): April 8, 2015
Received by editor(s) in revised form: May 18, 2015
Published electronically: September 15, 2015
Additional Notes: The work of the second author was supported by the ANR-12-BS01-0014-01 Geometrya.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society

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