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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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