Besicovitch Covering Property on graded groups and applications to measure differentiation

TL;DR

This paper characterizes when homogeneous and graded groups admit distances satisfying the Besicovitch Covering Property (BCP), with implications for measure differentiation, showing BCP holds only in certain low-step groups and not for sub-Riemannian distances.

Contribution

It provides a complete characterization of homogeneous groups with distances satisfying BCP, linking group structure to measure differentiation properties.

Findings

BCP holds in step 1 or 2 stratified groups

Homogeneous quasi-distances satisfy BCP iff layers commute

Sub-Riemannian distances on higher-step groups do not satisfy BCP

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.