Dichotomy of global density of Riesz capacity

• Autorzy: Hiroaki Aikawa
• Języki publikacji: EN

Abstrakty

EN

Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density
$𝓓̲(C_{α},E,r) = inf_{x∈ ℝⁿ} C_{α}(E∩B(x,r))/C_{α}(B(x,r))$
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 $lim_{r→ ∞} 𝓓̲ (C_{α},E,r)$ is either 0 or 1; the first case occurs if and only if $𝓓̲ (C_{α},E,r)$ 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.

Kategorie tematyczne

• 31B15: Potentials and capacities, extremal length

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2016
• Tom: 232
• Numer: 3
• Strony: 267-278

Daty

wydano

2016

Twórcy

autor

Hiroaki Aikawa

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

Identyfikatory

DOI

10.4064/sm8511-4-2016