Regularity versus smoothness of measures

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.

17 pages, 4 figures

References in corpus (3)

• Lower Assouad Dimension of Measures and Regularity
• Intermediate Assouad-like dimensions for measures
• Interpolating between dimensions