Real Analysis Exchange

Bloch and gap subharmonic functions.

R. Supper

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For subharmonic functions \(u\geq 0\) in the unit ball \(B_N\) of \(\mathbb{R}^N \), the paper characterizes this kind of growth: \(\sup_{x\in B_N} (1- \vert x\vert ^2 )^\alpha u(x) <+\infty \) (given \(\alpha >0\)), through criteria involving such integrals as \(\int u(x)\, dx \) or \(\int u(x) ( 1- \vert x\vert ^2 )^{\alpha -N} \, dx \) over balls centered at points \(a\in B_N\). Given \(p \in \mathbb{R}\) and \(\omega\) some non--negative function, this article compares subharmonic functions with the previous kind of growth to subharmonic functions satisfying: \( \sup_{a\in B_N} \int_{ B_N } u(x) ( 1- \vert x\vert ^2 )^p \omega (\vert \varphi _a (x)\vert)\, dx <+\infty \), where \(\varphi _a\) are Möbius transformations. The paper also studies subharmonic functions which are sums of lacunary series and their links with both previous kinds of subharmonic functions.

Article information

Real Anal. Exchange, Volume 28, Number 2 (2002), 395-414.

First available in Project Euclid: 20 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31B05: Harmonic, subharmonic, superharmonic functions 30B10: Power series (including lacunary series) 26B10: Implicit function theorems, Jacobians, transformations with several variables 26D15: Inequalities for sums, series and integrals 30D45: Bloch functions, normal functions, normal families

subharmonic function unit ball ellipsoid M\" obius transformation gap series


Supper, R. Bloch and gap subharmonic functions. Real Anal. Exchange 28 (2002), no. 2, 395--414.

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