## Real Analysis Exchange

### Bloch and gap subharmonic functions.

R. Supper

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 20 July 2007

https://projecteuclid.org/euclid.rae/1184963803

Mathematical Reviews number (MathSciNet)
MR2009762

Zentralblatt MATH identifier
1056.31003

#### Citation

Supper, R. Bloch and gap subharmonic functions. Real Anal. Exchange 28 (2002), no. 2, 395--414. https://projecteuclid.org/euclid.rae/1184963803

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