Local conical dimensions for measures

TL;DR

This paper investigates how measures decay within cones at small scales in Euclidean space, providing conditions to compute local dimensions via cones almost everywhere.

Contribution

It introduces new criteria for calculating local dimensions of measures using cone-based decay properties in Euclidean space.

Findings

Established decay rates of measures within cones

Provided conditions for local dimension calculation via cones

Applied results to various measure types in Euclidean space

Abstract

We study the decay of $\mu(B(x,r)\cap C)/\mu(B(x,r))$ as $r\downarrow 0$ for different kinds of measures $\mu$ on $\R^n$ and various cones $C$ around $x$. As an application, we provide sufficient conditions implying that the local dimensions can be calculated via cones almost everywhere.