Boundary behavior of \alpha-harmonic functions on the complement of the sphere and hyperplane

TL;DR

This paper investigates the boundary behavior of lpha-harmonic functions on the complements of spheres and hyperplanes, providing explicit formulas and establishing the Fatou theorem for these functions in Euclidean spaces.

Contribution

It offers new explicit formulas for Martin kernels, harmonic measures, and Green functions, and extends boundary behavior results to lpha-harmonic functions in these geometries.

Findings

Explicit formulas for Martin kernels and Green functions.

Proof of the Fatou theorem for lpha-harmonic functions.

Description of Hardy spaces on the complements.

Abstract

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\in(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\'evy processes. Some extensions for the relativistic \alpha-stable processes are discussed.