Entire $s$-harmonic functions are affine

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