Boundary oscillations of harmonic functions in Lipschitz domains

TL;DR

This paper extends the Law of Iterated Logarithm for harmonic functions with boundary growth constraints from halfspaces to Lipschitz domains, using Bloch function approximations to analyze boundary oscillations.

Contribution

It generalizes the LIL for harmonic functions to Lipschitz domains and introduces a Bloch function approximation method based on boundary growth conditions.

Findings

LIL holds for harmonic functions in Lipschitz domains.

Growth rate of Bloch functions depends on the weight w.

Slower growth rates than classical LIL for slowly increasing weights.

Abstract

Let $u(x,y)$ be a harmonic function in the halfspace $\mathbb{R}^n\times\mathbb{R}_+$ that grows near the boundary not faster than some fixed majorant $w(y)$. Recently it was proven that an appropriate weighted average along the vertical lines of such a function satisfies the Law of Iterated Logarithm (LIL). We extend this result to a class of Lipschitz domains in $\mathbb{R}^{n+1}$. In particular, we obtain the local version of this LIL for the upper halfspace. The proof is based on approximation of the weighted averages by a Bloch function, satisfying some additional condition determined by the weight $w$. The growth rate of such Bloch function depends on $w$ and, for slowly increasing $w$, turns out to be slower than the one provided by LILs of Makarov and Llorente. We discuss the necessary condition for an arbitrary Bloch function to exhibit this type of behaviour.