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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
MathSciNet review: 4002552
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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 \begin{equation*} \mu \big ( B(x, R) \setminus B(x,r) \big ) \leq C \frac {R-r}{R} \mu (B(x,R)) \end{equation*} 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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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
MR Author ID: 327617
Email: jgllorente@mat.uab.cat

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