Some Liouville theorems for the fractional Laplacian

In this paper, we prove the following result. Let $\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$\{\begin{array}{l}(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}u(x)=0,x\in {\mathbb{R}}^n,\\ {\displaystyle \frac{u(x)}{\mathrm{\mid }x{\mathrm{\mid }}^{\gamma }}\ge 0,}\end{array}$$ for some $0 \leq \gamma \leq 1$and$\gamma < \alpha$\mathrm{.}Then$u$mustbeconstantthroughout$$\mathbb{R}$^n$\mathrm{.}ThisisaLiouvilleTheoremfor$\alpha$-harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using potential theory. The other is by Fourier analysis as a corollary of the fact that the only $\alpha$-harmonic functions are affine.