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

Abstract

Let $M\colon (0,1) \to [e,+\infty)$ be a decreasing function such that $\int\limits_{0}^{1}\log\log M(y)dy<+\infty$. 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 $|u(x,y)| \leq M(|y|)$. We prove that $H_M$ is a normal family in $P$.