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

We study alpha-harmonic functions on the complement of the sphere and on the complement of the hyperplane in Euclidean spaces of dimension bigger than one, for alpha aaEuro parts per thousand(1,2). We describe the corresponding Hardy spaces and prove the Fatou theorem for alpha-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 alpha-stable L,vy processes. Some extensions for the relativistic alpha-stable processes are discussed.