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 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.
Citation
Fraser , J & Troscheit , S 2021 , ' Regularity versus smoothness of measures ' , Pacific Journal of Mathematics , vol. 311 , no. 2 , pp. 257–275 . https://doi.org/10.2140/pjm.2021.311.257
Publication
Pacific Journal of Mathematics
Status
Peer reviewed
ISSN
0030-8730Type
Journal article
Description
Funding: JMF was financially supported by the EPSRC Standard Grant EP/R015104/1 and the Leverhulme Trust Research Project Grant RPG-2019-034. ST was initially supported by the AÖU Collaborative Grant 103öu6 [in part] and later by the Austrian Science Fund (FWF) Meitner Fellowship M-2813.Collections
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