Regularity versus smoothness of measures

Jonathan Fraser, Sascha Troscheit

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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 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.
Original languageEnglish
Pages (from-to)257–275
Number of pages17
JournalPacific Journal of Mathematics
Issue number2
Publication statusPublished - 31 Jul 2021


  • Assouad dimension
  • Assouad spectrum
  • Lower dimension
  • Lower spectrum
  • Lp-spaces


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