JIPAM logo: Home Link
 
Home Editors Submissions Reviews Volumes RGMIA About Us
 

   
  Volume 2, Issue 2, Article 16
 
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 $0<B'<2B$).

         
       
  Download Screen PDF
  Download Print PDF
  Send this article to a friend
  Print this page
  

      search [advanced search] copyright 2003 terms and conditions login