On periodic $p$-harmonic functions on Cayley tree

TL;DR

This paper investigates the properties of periodic p-harmonic functions on Cayley trees, showing that those with finite index normal subgroup periodicity are constant, and describing non-constant classes for infinite index subgroups, highlighting non-linearity for p ≠ 2.

Contribution

It characterizes periodic p-harmonic functions on Cayley trees, proving constancy for finite index subgroups and constructing non-constant examples for infinite index, including linear combinations.

Findings

Finite index normal subgroup periodic p-harmonic functions are constant.

Non-constant periodic p-harmonic functions exist for some infinite index subgroups.

Linear combinations of certain p-harmonic functions remain p-harmonic despite non-linearity for p ≠ 2.

Abstract

We show that any periodic with respect to normal subgroups (of the group representation of the Cayley tree) of finite index $p$-harmonic function is a constant. For some normal subgroups of infinite index we describe a class of (non-constant) periodic $p$-harmonic functions. If $p\neq2$, the $p$-harmonicity is non-linear, i.e., the linear combination of $p$-harmonic functions need not be $p$-harmonic. In spite of this, we show that linear combinations of the $p$-harmonic functions described for normal subgroups of infinite index are also $p$-harmonic.