Regularity versus smoothness of measures

Jonathan M. Fraser and Sascha Troscheit

Vol. 311 (2021), No. 2, 257–275

DOI: 10.2140/pjm.2021.311.257

Abstract

The Assouad and lower dimensions and dimension spectra quantify the regularity of a measure by considering the relative measure of concentric balls. On the other hand, one can quantify the smoothness of an absolutely continuous measure by considering the $L^p$ norms of its density. We establish sharp relationships between these two notions. Roughly speaking, we show that smooth measures must be regular, but that regular measures need not be smooth.

Keywords

smoothness of measures, regularity of measures, Assouad dimension

Mathematical Subject Classification

Primary: 28A80

Secondary: 37C45

Milestones

Received: 19 March 2020

Accepted: 14 January 2021

Published: 31 July 2021