#### Vol. 311, No. 2, 2021

 Recent Issues Vol. 315: 1  2 Vol. 314: 1  2 Vol. 313: 1  2 Vol. 312: 1  2 Vol. 311: 1  2 Vol. 310: 1  2 Vol. 309: 1  2 Vol. 308: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
Regularity versus smoothness of measures

### Jonathan M. Fraser and Sascha Troscheit

Vol. 311 (2021), No. 2, 257–275
##### 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.

##### Keywords
smoothness of measures, regularity of measures, Assouad dimension
Primary: 28A80
Secondary: 37C45