Liouville Theorem Involving the Uniformly Nonlocal Operator

We prove that u is constant if u is a bounded solution of A α u ( x ) = C n , α P.V. ∫ R n a ( x - y ) ( u ( x ) - u ( y ) ) | x - y | n + α d y = 0 , x ∈ R n , where the function a : R n ↦ R be uniformly bounded and radial decreasing. This result can be regarded as the generalization of usual Liouville theorem. To get the proof, we establish a maximum principle involving the nonlocal operator A α for antisymmetric functions on any half space.