Radial growth of harmonic functions in the unit ball

TL;DR

This paper investigates the growth behavior of harmonic functions in the unit ball, showing that under a doubling condition on the majorant, extremal growth occurs only on small sets of radii with precise estimates.

Contribution

It establishes a link between doubling conditions on the majorant and the restricted growth or decay of harmonic functions in the unit ball, providing detailed estimates of exceptional sets.

Findings

Extremal growth occurs only on small sets of radii.

Precise estimates of exceptional sets are provided.

Doubling condition on the majorant influences growth behavior.

Abstract

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give precise estimates of these exceptional sets.