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Regularity versus
smoothness of measures
Jonathan M. Fraser and Sascha Troscheit |
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Vol. 311 (2021), No. 2, 257–275
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Abstract |
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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 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
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Mathematical Subject Classification
Primary: 28A80
Secondary: 37C45
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Milestones
Received: 19 March 2020
Accepted: 14 January 2021
Published: 31 July 2021
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Authors |
| Jonathan M. Fraser | |
| School of Mathematics and
Statistics University of St Andrews St Andrews United Kingdom |
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| Sascha Troscheit | |
| Faculty of Mathematics University of Vienna Wien Austria |
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