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

TL;DR

This paper extends the Levinson log log theorem to higher dimensions, proving that a family of harmonic functions with controlled growth near the boundary is normal, thus generalizing classical complex analysis results.

Contribution

It introduces a higher-dimensional harmonic analog of the Levinson log log theorem, establishing normality of families of harmonic functions with boundary growth constraints.

Findings

Proves the normality of harmonic function families under growth conditions.

Generalizes the Levinson log log theorem to higher dimensions.

Provides a framework for boundary behavior of harmonic functions.

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 $P$.