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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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.
- [1] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions, Proc. Cambridge Philos. Soc. 41 (1945), 103–110. MR 12325, https://doi.org/10.1017/s0305004100022453
- [2] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions. II, Proc. Cambridge Philos. Soc. 42 (1946), 1–10. MR 14414, https://doi.org/10.1017/s0305004100022660
- [3] Waldemar Hebisch and Adam Sikora, A smooth subadditive homogeneous norm on a homogeneous group, Studia Math. 96 (1990), no. 3, 231–236. MR 1067309, https://doi.org/10.4064/sm-96-3-231-236
- [4] Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917
- [5] A. Korányi and H. M. Reimann, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), no. 1, 1–87. MR 1317384, https://doi.org/10.1006/aima.1995.1017
- [6] E. Le Donne, S. Rigot, Besicovitch Covering Property for homogeneous distances in the Heisenberg groups, preprint, http://arxiv.org/abs/1406.1484
- [7] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
- [8] Séverine Rigot, Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carathéodory metric, Math. Z. 248 (2004), no. 4, 827–848. MR 2103544, https://doi.org/10.1007/s00209-004-0683-7
- [9] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. MR 1175693, https://doi.org/10.2307/2374799
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 28C15, 49Q15, 43A80
Retrieve articles in all journals with MSC (2010): 28C15, 49Q15, 43A80
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
