Dichotomy of global capacity density
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- by Hiroaki Aikawa and Tsubasa Itoh PDF
- Proc. Amer. Math. Soc. 143 (2015), 5381-5393 Request permission
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 \[ \lim _{r\to \infty }\Bigg (\inf _{x\in \mathbb {R}^n} \frac {\mathrm {Cap}_{p,\mu }(E\cap B(x,r),B(x,\lambda r))}{\mathrm {Cap}_{p,\mu }(B(x,r),B(x,\lambda r))} \Bigg )\] is equal to either 0 or 1; the first case occurs if and only if \[ \inf _{x\in \mathbb {R}^n} \frac {\mathrm {Cap}_{p,\mu }(E\cap B(x,r_0), B(x,\lambda r))}{\mathrm {Cap}_{p,\mu }(B(x,r), B(x,\lambda r))} \] is identically equal to 0. This provides a sharp contrast between capacity and Lebesgue measure.References
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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
- MR Author ID: 997062
- Email: tsubasa@math.titech.ac.jp
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5381-5393
- MSC (2010): Primary 31C15
- DOI: https://doi.org/10.1090/proc/12672
- MathSciNet review: 3411153