On the higher-dimensional harmonic analog of the Levinson log log theorem

Abstract

Let $M:(0,1)\to [e,+\infty )$ be a decreasing function such that $\underset{0}{\overset{1}{\int }}\log \log M(y)dy<+\infty$. Consider the set $H_M$ of all functions $u$ harmonic in $P:=\{(x,y)\in {\mathbb{R}}^n:x\in {\mathbb{R}}^{n-1},y\in \mathbb{R},|x|<1,|y|<1\}$ and satisfying $|u(x,y)|\le M(|y|)$. We prove that $H_M$ is a normal family in $P$.