Wavelet decomposition of harmonic functions in growth spaces

TL;DR

This paper explores how harmonic functions in the upper half-space can be analyzed using wavelet decomposition, leading to new insights into their boundary behavior and oscillation properties.

Contribution

It introduces a wavelet-based framework for describing growth spaces of harmonic functions and applies it to establish a law of the iterated logarithm for their oscillations.

Findings

Wavelet decomposition characterizes harmonic functions in growth spaces.

Law of the iterated logarithm is proved for harmonic oscillations.

Multiresolution approximations effectively describe boundary behavior.

Abstract

Spaces of harmonic functions in upper half-space with controlled growth near the boundary are described in terms of multiresolution approximations. The results are applied to prove the law of the iterated logarithm for the oscillation of harmonic functions along vertical lines.