Regularity versus smoothness of measures

TL;DR

This paper explores the relationship between measure regularity, as quantified by Assouad and lower dimensions, and measure smoothness, characterized by $L^p$ norms, establishing that smoothness implies regularity but not vice versa.

Contribution

The paper establishes sharp relationships between measure regularity and smoothness, clarifying how these concepts are interconnected and highlighting their differences.

Findings

Smooth measures are necessarily regular.

Regular measures are not necessarily smooth.

Sharp bounds linking regularity and smoothness are derived.

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.