Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and Hyperplane

Abstract

We study α-harmonic functions on the complement of the sphere and on the complement of the hyperplane in Euclidean spaces of dimension bigger than one, for α ∈ (1,2). We describe the corresponding Hardy spaces and prove the Fatou theorem for α-harmonic functions. We also give explicit formulas for the Martin kernel of the complement of the sphere and for the harmonic measure, Green function and Martin kernel of the complement of the hyperplane for the symmetric α-stable Levy processes. Some extensions for the relativistic α-stable processes are discussed.