JIPAM

Subharmonic Functions and their Riesz Measure  
 
  Authors: Raphaele Supper,  
  Keywords: Subharmonic functions, Order of growth, Riesz measure  
  Date Received: 30/08/00  
  Date Accepted: 22/01/01  
  Subject Codes:

31A05,31B05,26D15,28A75

  
  Editors: Alberto Fiorenza,  
 
  Abstract:

For subharmonic functions $u$ in ${ I!! R}^N$, of Riesz measure $mu$, the growth of the function $s mapsto mu (s) =int_{vert zeta vert leq s } dmu (zeta) $ ($sgeq 0$) is described and compared with the growth of $u$. It is also shown that, if $ int_{{ I!! R}^N} u^+ (x) [-varphi ' (vert xvert ^2)] dx <+infty$ for some decreasing $ C ^1$ function $varphi geq 0$, then $ int_{{ I!! R}^N} {1over{vert zeta vert ^2 }} varphi(vert zeta vert ^2 +1) dmu (zeta ) <+infty .$ Given two subharmonic functions $u_1$ and $u_2$, of Riesz measures $mu_1$ and $mu_2$, with a growth like $u_i (x)leq A+ B vert xvert ^gamma$ $forall x in { I!! R}^N$ ($i=1,2$), it is proved that $mu_1 +mu_2$ is not necessarily the Riesz measure of any subharmonic function $u$ with such a growth as $u (x)leq A'+ B' vert xvert ^gamma$$forall x in { I!! R}^N$ (here $A>0$, $A'>0$ and $0<B'<2B$).
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