A Fatou-type Theorem for Harmonic Functions on Symmetric Spaces1 by S. Helgason and A. Koranyi

A Fatou-type Theorem for Harmonic Functions on Symmetric Spaces

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

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 an...

Ja n 20 04 Relative Fatou ’ s Theorem for ( − ∆ ) α / 2 - harmonic Functions in Bounded κ - fat Open Set ∗

Recently it was shown in Kim [26] that Fatou's theorem for transient censored α-stable processes in a bounded C 1,1 open set is true. Here we give a probabilistic proof of relative Fatou's theorem for (−∆) α/2-harmonic functions (equivalently for symmetric α-stable processes) in bounded κ-fat open set where α ∈ (0, 2). That is, if u is positive (−∆) α/2-harmonic function in a bounded κ-fat open...

Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any α ∈ (0, 2), a truncated symmetric α-stable process in R is a symmetric Lévy process in R with no diffusion part and with a Lévy density given by c|x| 1{|x|<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this proc...

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let L = ∆α/2 + b · ∇ with α ∈ (1, 2). We prove the Martin representation and the Relative Fatou Theorem for non-negative singular L-harmonic functions on C1,1 bounded open sets. 1