Vol. 311, No. 2, 2021

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Regularity versus smoothness of measures

Jonathan M. Fraser and Sascha Troscheit

Vol. 311 (2021), No. 2, 257–275
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 Lp 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.

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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
Authors
Jonathan M. Fraser
School of Mathematics and Statistics
University of St Andrews
St Andrews
United Kingdom
Sascha Troscheit
Faculty of Mathematics
University of Vienna
Wien
Austria