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Subharmonic Functions in the Unit Ball

Authors:	Raphaële Supper

Institution:	(1) UFR de Mathématique et Informatique, URA CNRS 001, Université Louis Pasteur, 7 rue René Descartes, F–67 084, Strasbourg Cedex, France

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

Keywords:	subharmonic functions growth Riesz measure unit ball
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