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arXiv · 1912.07292

Regularity versus smoothness of measures

Jonathan M. Fraser·Sascha Troscheit
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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.

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Jonathan M. Fraser, Sascha Troscheit. 2019-12-16. Regularity versus smoothness of measures. https://doi.org/10.2140/pjm.2021.311.257

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