The Besicovitch covering property in the Heisenberg group revisited

TL;DR

This paper investigates the Besicovitch covering property in the Heisenberg group, providing criteria, characterizations, and new classes of distances where BCP holds, enhancing understanding in geometric measure theory.

Contribution

It offers necessary and sufficient conditions for BCP in the Heisenberg group with homogeneous distances, including characterizations for rotationally invariant cases.

Findings

Criteria for BCP validity in the Heisenberg group

New classes of homogeneous distances satisfying BCP

Full characterization of rotationally invariant distances with BCP

Abstract

The Besicovitch covering property (BCP) is known to be one of the fundamental tools in measure theory, and more generally, a usefull 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.