Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator

TL;DR

This paper constructs a harmonic function in Euclidean space that is frequently hypercyclic for a partial differentiation operator and exhibits minimal growth in average $L^2$-norm, solving a problem posed in 2010.

Contribution

It introduces a harmonic function with minimal growth that is frequently hypercyclic for a partial differentiation operator, addressing an open problem from 2010.

Findings

Constructed a harmonic function with minimal growth rate.

Proved the function is frequently hypercyclic for the partial differentiation operator.

Solved a problem posed by Blasco, Bonilla, and Grosse-Erdmann in 2010.

Abstract

We solve a problem posed by Blasco, Bonilla and Grosse-Erdmann in 2010 by constructing a harmonic function on $\mathbb{R}^N$, that is frequently hypercyclic with respect to the partial differentiation operator $\partial/\partial x_k$ and which has a minimal growth rate in terms of the average $L^2$-norm on spheres of radius $r>0$ as $r \to \infty$.