Abstract
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.
Citation
R. Supper. "Bloch and gap subharmonic functions.." Real Anal. Exchange 28 (2) 395 - 414, 2002/2003.
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