Dichotomy of global density of Riesz capacity

Hiroaki Aikawa

Hiroaki Aikawa

Studia Mathematica, Tome 233 (2016), p. 267-278 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲ (C_{α}, E, r) = i n f_{x \in ℝ ⁿ} C_{α} (E \cap B (x, r)) / C_{α} (B (x, r))$ 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 $l i m_{r \to \infty} ̲ (C_{α}, E, r)$ 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.

Publié le : 2016-01-01

Zbl 06586863

EUDML-ID : urn:eudml:doc:286193

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8511-4-2016,
     author = {Hiroaki Aikawa},
     title = {Dichotomy of global density of Riesz capacity},
     journal = {Studia Mathematica},
     volume = {233},
     year = {2016},
     pages = {267-278},
     zbl = {06586863},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8511-4-2016}
}

Hiroaki Aikawa. Dichotomy of global density of Riesz capacity. Studia Mathematica, Tome 233 (2016) pp. 267-278. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8511-4-2016/