Besicovitch Covering Property on graded groups and applications to measure differentiation
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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.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-12-BS01-0014-01
Award Identifier / Grant number: ANR-15-CE40-0018
Funding source: Suomen Akatemia
Award Identifier / Grant number: 288501
Funding statement: The first-named author acknowledges the support of the Academy of Finland project no. 288501. The second-named author is partially supported by ANR grants ANR-12-BS01-0014-01 and ANR-15-CE40-0018.
Acknowledgements
The authors would like to thank Tapio Rajala for fruitful conversations and improving feedback.
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© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Supercuspidal representations and preservation principle of theta correspondence
- Essential regularity of the model space for the Weil–Petersson metric
- Generalized Lagrangian mean curvature flows: The cotangent bundle case
- Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties
- Imaginaries and invariant types in existentially closed valued differential fields
- Representations of categories of G-maps
- Sharply 2-transitive groups of characteristic 0
- Addendum to Sharply 2-transitive groups of characteristic 0
- Besicovitch Covering Property on graded groups and applications to measure differentiation
Artikel in diesem Heft
- Frontmatter
- Supercuspidal representations and preservation principle of theta correspondence
- Essential regularity of the model space for the Weil–Petersson metric
- Generalized Lagrangian mean curvature flows: The cotangent bundle case
- Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties
- Imaginaries and invariant types in existentially closed valued differential fields
- Representations of categories of G-maps
- Sharply 2-transitive groups of characteristic 0
- Addendum to Sharply 2-transitive groups of characteristic 0
- Besicovitch Covering Property on graded groups and applications to measure differentiation
