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On a class of singular measures satisfying a strong annular decay condition


Authors: Ángel Arroyo and José G. Llorente
Journal: Proc. Amer. Math. Soc. 147 (2019), 4409-4423
MSC (2010): Primary 28A75, 30L99
DOI: https://doi.org/10.1090/proc/14576
Published electronically: May 17, 2019
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Abstract: A metric measure space $ (X, d, \mu )$ is said to satisfy the strong annular decay condition if there is a constant $ C>0$ such that

$\displaystyle \mu \big ( B(x, R) \setminus B(x,r) \big ) \leq C\, \frac {R-r}{R}\, \mu (B(x,R))$    

for each $ x\in X$ and all $ 0<r \leq R$. If $ d_{\infty }$ is the distance induced by the $ \infty $-norm in $ \mathbb{R}^N$, we construct examples of singular measures $ \mu $ on $ \mathbb{R}^N$ such that $ (\mathbb{R}^N, d_{\infty }, \mu )$ satisfies the strong annular decay condition.

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Additional Information

Ángel Arroyo
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
Email: angel.a.arroyo@jyu.fi

José G. Llorente
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: jgllorente@mat.uab.cat

DOI: https://doi.org/10.1090/proc/14576
Keywords: Annular decay condition, doubling measure, Bernoulli product, metric measure space
Received by editor(s): September 4, 2018
Received by editor(s) in revised form: January 25, 2019
Published electronically: May 17, 2019
Additional Notes: This research was partially supported by grants MTM2017-85666-P, 2017 SGR 395.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2019 American Mathematical Society