Bloch and gap subharmonic functions.

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.