Dichotomy of global density of Riesz capacity

Hiroaki Aikawa¹

¹ Department of Mathematics Hokkaido University Sapporo 060-0810, Japan

Studia Mathematica, Tome 232 (2016) no. 3, pp. 267-278

Résumé

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.

Détail

DOI : 10.4064/sm8511-4-2016

Keywords: alpha riesz capacity order alpha alpha mathbb consider riesz capacity density underline mathcal alpha operatorname inf mathbb frac alpha cap alpha borel set subset mathbb where stands ball center radius nbsp alpha lim infty underline mathcal alpha either first occurs only underline mathcal alpha identically zero moreover shown densities respect general sets enjoy dichotomy nbsp decay estimate alpha capacitary potentials obtained