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 {(-δ)α/2u(x)=0,x∈Rn,lim¯|x|→∞u(x)|x|γ≤0, for some 0≥≥;1 and γα. Then u must be constant throughoutRn. 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.