A Liouville Theorem for the Fractional Laplacian
  We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove   Every $\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant. 
Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan
math.AP