Subharmonic Functions in the Unit Ball

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 $\int_{{B}_{N}} h(\sqrt{|\zeta|}) \omega(\sqrt{|\zeta|}) d\mu (\zeta)< +\infty$.