Absence of non-constant harmonic functions with $\ell^p$-gradient in some semi-direct products

TL;DR

This paper investigates semi-direct product groups and demonstrates that many do not support harmonic functions with gradients in ^p, highlighting limitations in the existence of such functions in these structures.

Contribution

It establishes that numerous semi-direct product groups lack harmonic functions with ^p gradients, extending understanding of harmonic function behavior in these groups.

Findings

Many semi-direct product groups do not admit harmonic functions with ^p gradients.

The results apply to groups like lamplighter groups and similar structures.

This work narrows the class of groups supporting such harmonic functions.

Abstract

To obtain groups with bounded harmonic functions (which are not hyperbolic), one of the most frequent way is to look at some semi-direct products (\eg lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in $\ell^p$, for $p\in [1,\infty[$.