Dichotomy of global density of Riesz capacity
Volume 232 / 2016
Abstract
Let $C_\alpha $ be the Riesz capacity of order $\alpha $, $0 \lt \alpha \lt n$, in ${{\mathbb R}^n}$. We consider the Riesz capacity density $$ \underline {\mathcal {D}}(C_\alpha ,E,r)=\operatorname {inf}_{x\in {{\mathbb R}^n}}\frac {C_\alpha (E\cap B(x,r))}{C_\alpha (B(x,r))} $$ for a Borel set $E\subset {{\mathbb R}^n}$, where $B(x,r)$ stands for the open ball with center at $x$ and radius $r$. In case $0 \lt \alpha \le 2$, we show that $\lim_{r\to \infty }\underline {\mathcal {D}} (C_\alpha ,E,r)$ is either 0 or 1; the first case occurs if and only if $\underline {\mathcal {D}} (C_\alpha ,E,r)$ is identically zero for all $r \gt 0$. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for $\alpha $-capacitary potentials is also obtained.
