Mean values and associated measures of δ -subharmonic functions

Neil A. Watson

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 1, page 83-102
  • ISSN: 0862-7959

Abstract

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Let u be a δ -subharmonic function with associated measure μ , and let v be a superharmonic function with associated measure ν , on an open set E . For any closed ball B ( x , r ) , of centre x and radius r , contained in E , let ( 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 0 with 0 < s < t of the quotient ( ( u , x , s ) - ( u , x , t ) ) / ( ( v , x , s ) - ( v , x , t ) ) , lie between the upper and lower limits as r 0 + of the quotient μ ( B ( x , r ) ) / ν ( B ( x , r ) ) . This enables us to use some well-known measure-theoretic results to prove new variants and generalizations of several theorems about δ -subharmonic functions.

How to cite

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Watson, Neil A.. "Mean values and associated measures of $\delta $-subharmonic functions." Mathematica Bohemica 127.1 (2002): 83-102. <http://eudml.org/doc/249014>.

@article{Watson2002,
abstract = {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.},
author = {Watson, Neil A.},
journal = {Mathematica Bohemica},
keywords = {superharmonic; $\delta $-subharmonic; Riesz measure; spherical mean values; superharmonic function; -subharmonic function; Riesz measure; spherical mean values},
language = {eng},
number = {1},
pages = {83-102},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Mean values and associated measures of $\delta $-subharmonic functions},
url = {http://eudml.org/doc/249014},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Watson, Neil A.
TI - Mean values and associated measures of $\delta $-subharmonic functions
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 1
SP - 83
EP - 102
AB - 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.
LA - eng
KW - superharmonic; $\delta $-subharmonic; Riesz measure; spherical mean values; superharmonic function; -subharmonic function; Riesz measure; spherical mean values
UR - http://eudml.org/doc/249014
ER -

References

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