Entire $s$-harmonic functions are affine

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.

The proofs has been changed and are now based on Cauchy-type estimates for harmonic functions. To appear in Proceedings of the AMS