Some Liouville theorems for the fractional Laplacian

In this paper, we prove the following result. Let α be any real number between 0 and 2. Assume that u is a solution of {(−△)α/2u(x)=0,x∈Rn,lim¯∣x∣→∞u(x)∣x∣γ≥0, for some 0≤γ≤1 and γ<α. Then u must be constant throughout Rn. This is a Liouville Theorem for α -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 α -harmonic functions are affine.