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.
Categories: math.AP