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

TL;DR

This paper constructs examples of singular measures on Euclidean space with the infinity norm that satisfy a strong annular decay condition, expanding understanding of measure behaviors in metric measure spaces.

Contribution

The paper introduces specific singular measures on rica^N that satisfy the strong annular decay condition, providing new examples in metric measure space theory.

Findings

Existence of singular measures satisfying the condition

Construction of measures on rica^N with the inity-norm

Insights into measure decay properties in metric spaces

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