  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.

Analysis of PDEs