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

Arroyo, Ángel; Llorente, José

doi:10.1090/proc/14576

On a class of singular measures satisfying a strong annular decay condition
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by Ángel Arroyo and José G. Llorente PDF

Proc. Amer. Math. Soc. 147 (2019), 4409-4423 Request permission

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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