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Title: Mean values and associated measures of $\delta $-subharmonic functions (English)
Author: Watson, Neil A.
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 127
Issue: 1
Year: 2002
Pages: 83-102
Summary lang: English
Category: math
Summary: Let $u$ be a $\delta $-subharmonic function with associated measure $\mu $, and let $v$ be a superharmonic function with associated measure $\nu $, on an open set $E$. For any closed ball $B(x,r)$, of centre $x$ and radius $r$, contained in $E$, let ${\mathcal M}(u,x,r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s,t\rightarrow 0$ with $0<s<t$ of the quotient $({\mathcal M}(u,x,s)-{\mathcal M}(u,x,t))/({\mathcal M}(v,x,s)-{\mathcal M}(v,x,t))$, lie between the upper and lower limits as $r\rightarrow 0+$ of the quotient $\mu (B(x,r))/\nu (B(x,r))$. This enables us to use some well-known measure-theoretic results to prove new variants and generalizations of several theorems about $\delta $-subharmonic functions. (English)
Keyword: superharmonic
Keyword: $\delta $-subharmonic
Keyword: Riesz measure
Keyword: spherical mean values
MSC: 31B05
idZBL: Zbl 0998.31002
idMR: MR1895249
DOI: 10.21136/MB.2002.133981
Date available: 2009-09-24T21:58:22Z
Last updated: 2020-07-29
Stable URL:
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