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Subharmonic Functions in the Unit Ball

Authors:	Raphaële Supper

Affiliation:	(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 , 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 $int_{{B}_{N}} h(sqrt{\|zeta\|}) omega(sqrt{\|zeta\|}) dmu (zeta)< +infty$ .

Keywords:	subharmonic functions growth Riesz measure unit ball
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