Dichotomy of global density of Riesz capacity

Hiroaki Aikawa. "Dichotomy of global density of Riesz capacity." Studia Mathematica 232.3 (2016): 267-278. <http://eudml.org/doc/286193>.

@article{HiroakiAikawa2016,
abstract = {Let $C_\{α\}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲(C_\{α\},E,r) = inf_\{x∈ ℝⁿ\} C_\{α\}(E∩B(x,r))/C_\{α\}(B(x,r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $lim_\{r→ ∞\} ̲ (C_\{α\},E,r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_\{α\},E,r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also obtained.},
author = {Hiroaki Aikawa},
journal = {Studia Mathematica},
keywords = {dichotomy; Riesz capacity; density; fractional Laplacian; $\alpha $harmonic},
language = {eng},
number = {3},
pages = {267-278},
title = {Dichotomy of global density of Riesz capacity},
url = {http://eudml.org/doc/286193},
volume = {232},
year = {2016},
}

TY - JOUR
AU - Hiroaki Aikawa
TI - Dichotomy of global density of Riesz capacity
JO - Studia Mathematica
PY - 2016
VL - 232
IS - 3
SP - 267
EP - 278
AB - Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲(C_{α},E,r) = inf_{x∈ ℝⁿ} C_{α}(E∩B(x,r))/C_{α}(B(x,r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $lim_{r→ ∞} ̲ (C_{α},E,r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_{α},E,r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also obtained.
LA - eng
KW - dichotomy; Riesz capacity; density; fractional Laplacian; $\alpha $harmonic
UR - http://eudml.org/doc/286193
ER -