Regularity versus smoothness of measures

Jonathan Fraser; Sascha Troscheit

doi:10.2140/pjm.2021.311.257

Regularity versus smoothness of measures

Jonathan Fraser, Sascha Troscheit

Research output: Contribution to journal › Article › peer-review

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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.

Original language	English
Pages (from-to)	257–275
Number of pages	17
Journal	Pacific Journal of Mathematics
Volume	311
Issue number	2
DOIs	https://doi.org/10.2140/pjm.2021.311.257
Publication status	Published - 31 Jul 2021

Keywords

Assouad dimension
Assouad spectrum
Lower dimension
Lower spectrum
Lp-spaces

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10.2140/pjm.2021.311.257

Fraser-2021-Reg-v-smooth-PJM-AAM
Copyright © 2021 Pacific Journal of Mathematics. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.2140/pjm.2021.311.257.
Accepted author manuscript, 395 KB

https://arxiv.org/abs/1912.07292

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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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.",

keywords = "Assouad dimension, Assouad spectrum, Lower dimension, Lower spectrum, Lp-spaces",

author = "Jonathan Fraser and Sascha Troscheit",

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KW - Assouad dimension

KW - Assouad spectrum

KW - Lower dimension

KW - Lower spectrum

KW - Lp-spaces

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Regularity versus smoothness of measures

Abstract

Keywords

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Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this