# Subharmonic Functions in the Unit Ball

Subharmonic Functions in the Unit Ball For functions u subharmonic in the unit ball B N of $${\mathbb R}^N$$ , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B N . Cases under study are: $$u(x) \leq A+ B [ {h(\vert x \vert )}]^{-\gamma}$$ and $$u(x) \leq A+ B, h(1-\vert x \vert ),forall x in B_N$$ , with A, B, γ positive constants and $$h(s)=\log \frac{1}{s}$$ if N=2 or $$h(s)=\frac{1}{{s^{N-2}}}- 1$$ if N≥ 3. This paper contains several integral results, as for instance: when ∫BN u+(x)[-ω′(|x|2)]dx < +∞ for some positive decreasing C1 function ω, it is proved that $$\int_{{B}_{N}} h(\sqrt{|\zeta|}) \omega(\sqrt{|\zeta|}) d\mu (\zeta)< +\infty$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Subharmonic Functions in the Unit Ball

, Volume 9 (4) – Jan 1, 2005
21 pages

/lp/springer_journal/subharmonic-functions-in-the-unit-ball-Sn9rc0zIXM
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-005-2716-9
Publisher site
See Article on Publisher Site

### Abstract

For functions u subharmonic in the unit ball B N of $${\mathbb R}^N$$ , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B N . Cases under study are: $$u(x) \leq A+ B [ {h(\vert x \vert )}]^{-\gamma}$$ and $$u(x) \leq A+ B, h(1-\vert x \vert ),forall x in B_N$$ , with A, B, γ positive constants and $$h(s)=\log \frac{1}{s}$$ if N=2 or $$h(s)=\frac{1}{{s^{N-2}}}- 1$$ if N≥ 3. This paper contains several integral results, as for instance: when ∫BN u+(x)[-ω′(|x|2)]dx < +∞ for some positive decreasing C1 function ω, it is proved that $$\int_{{B}_{N}} h(\sqrt{|\zeta|}) \omega(\sqrt{|\zeta|}) d\mu (\zeta)< +\infty$$ .

### Journal

PositivitySpringer Journals

Published: Jan 1, 2005

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