6533b7d2fe1ef96bd125e26d

RESEARCH PRODUCT

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

ÁNgel ArroyoJosé G. Llorente

subject

PhysicsClass (set theory)Applied MathematicsGeneral MathematicsMetric Geometry (math.MG)Space (mathematics)metriset avaruudetMeasure (mathematics)Bernoulli productfunktioteoriaCombinatoricsmetric measure spaceMathematics - Metric Geometryannular decay conditiondoubling measureFOS: Mathematicsmittateoria

description

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.

http://arxiv.org/abs/1808.07669