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Dichotomy of global capacity density


Authors: Hiroaki Aikawa and Tsubasa Itoh
Journal: Proc. Amer. Math. Soc. 143 (2015), 5381-5393
MSC (2010): Primary 31C15
DOI: https://doi.org/10.1090/proc/12672
Published electronically: April 21, 2015
MathSciNet review: 3411153
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Abstract: Let $ 1<p<\infty $ and let $ d\mu (x)=w(x)dx$ be a $ p$-admissible weight in $ \mathbb{R}^n$, $ n\ge 2$. By $ \mathrm {Cap}_{p,\mu }(E,D)$ we denote the variational $ (p,\mu )$-capacity of condenser $ (E,D)$. We show a dichotomy of the global density with respect to $ \mathrm {Cap}_{p,\mu }$. One of our results is as follows: Let $ \lambda >1$ and let $ B(x,r)$ stand for the open ball with center at $ x$ and radius $ r$. Then

$\displaystyle \lim _{r\to \infty }\Bigg (\inf _{x\in \mathbb{R}^n} \frac {\math... ... B(x,r),B(x,\lambda r))}{\mathrm {Cap}_{p,\mu }(B(x,r),B(x,\lambda r))} \Bigg )$

is equal to either 0 or 1; the first case occurs if and only if

$\displaystyle \inf _{x\in \mathbb{R}^n} \frac {\mathrm {Cap}_{p,\mu }(E\cap B(x,r_0), B(x,\lambda r))}{\mathrm {Cap}_{p,\mu }(B(x,r), B(x,\lambda r))} $

is identically equal to 0. This provides a sharp contrast between capacity and Lebesgue measure.

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Additional Information

Hiroaki Aikawa
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Email: aik@math.sci.hokudai.ac.jp

Tsubasa Itoh
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama Meguro-ku Tokyo 152-8551, Japan
Email: tsubasa@math.titech.ac.jp

DOI: https://doi.org/10.1090/proc/12672
Keywords: Dichotomy, capacity, density, harmonic measure
Received by editor(s): August 4, 2014
Received by editor(s) in revised form: November 11, 2014
Published electronically: April 21, 2015
Additional Notes: The first named author was supported in part by JSPS KAKENHI Grant Numbers 25287015 and 25610017.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society