The Besicovitch covering property (BCP) is known to be one of the fundamental tools in measure theory, and more generally, a useful property for numerous purposes in analysis and geometry. We prove both sufficient and necessary criteria for the validity of BCP in the first Heisenberg group equipped with a homogeneous distance. Beyond recovering all previously known results about the validity or non-validity of BCP in this setting, we get simple descriptions of new large classes of homogeneous distances satisfying BCP. We also obtain a full characterization of rotationally invariant distances for which BCP holds in the first Heisenberg group under mild regularity assumptions about their unit sphere.
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S.G. has been supported by the European Unions Seventh Framework Programme, Marie Curie Actions-Initial Training Network, under Grant Agreement No. 607643, “Metric Analysis For Emergent Technologies (MAnET)”; and by the EPSRC Grant “Sub-Elliptic Harmonic Analysis” (EP/P002447/1).
S.R. is partially supported by ANR Grant ANR-15-CE40-0018.
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Golo, S., Rigot, S. The Besicovitch covering property in the Heisenberg group revisited. J Geom Anal 29, 3345–3383 (2019). https://doi.org/10.1007/s12220-018-00112-z
- Heisenberg group
- Covering theorems
- Homogeneous groups
Mathematics Subject Classification