, Volume 39, Issue 1, pp 29-67
Date: 10 Nov 2012

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

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 Lévy processes. Some extensions for the relativistic α-stable processes are discussed.

This research was partially supported by Agence Nationale de la Recherche grant ANR-09-BLAN-0084-01 and by MNiSW grant N N201 373136.