Entire $s$-harmonic functions are affine

TL;DR

This paper proves that all solutions to the fractional Laplace equation in the entire space are affine functions, leading to a uniqueness result for the Riesz potential in Lebesgue spaces.

Contribution

It establishes that entire $s$-harmonic functions are affine, providing a key uniqueness result for the Riesz potential in Lebesgue spaces.

Findings

All solutions to $(- riangle)^s u=0$ in $R^N$ are affine functions.

Proves the uniqueness of the Riesz potential $|x|^{2s-N}$ in Lebesgue spaces.

Enhances understanding of fractional harmonic functions and their properties.

Abstract

In this paper, we prove that solutions to the equation $(-\Delta)^s u=0$ in $\mathbb{R}^N$, for $s\in (0,1)$, are affine. This will allow us to prove uniqueness of the Riesz potential $|x|^{2s-N}$ in Lebesgue spaces.