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 $$\left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\ \displaystyle\underset{|x| \to \infty}{\underline{\lim}} \frac{u(x)}{|x|^{\gamma}} \geq 0 , \end{array} \right.$$ for some $0 \leq \gamma \leq 1$ and $\gamma < \alpha$. Then $u$ must be constant throughout $\mathbb{R}^n$. This is a Liouville Theorem for $\alpha$-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.