Scaling properties of Hausdorff and packing measures

Abstract.

Let $m\in{\mathbb N}$. Let $\theta$ be a continuous increasing function defined on ${\mathbb R}^+$, for which $\theta(0)=0$ and $\theta(t)/t^m$ is a decreasing function of t. Let $\|\cdot\|$ be a norm on ${\mathbb R}^m$, and let $\rho$, ${\mathcal H}^\theta ={\mathcal H}^\theta_\rho$, ${\mathcal P}^\theta ={\mathcal P}^\theta_\rho$ denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions $\theta$ such that the corresponding Hausdorff or packing measure scales with exponent $\alpha$ by showing it must be of the form $\theta(t) = t^\alpha L(t)$, where L is slowly varying. We also show that for continuous increasing functions $\theta$ and $\eta$ defined on ${\mathbb R}^+$, for which $\theta(0)=\eta(0)=0$, ${\mathcal H}^\theta={\mathcal P}^\eta$ is either trivially true or false: we show that if ${\mathcal H}^\theta={\mathcal P}^\eta$, then ${\mathcal H}^\theta={\mathcal P}^\eta=c\cdot\lambda$ for a constant c, where $\lambda$ is the Lebesgue measure on ${\mathbb R}^m$.