Fatou theorem of p-harmonic functions on trees

A Fatou-type Theorem for Harmonic Functions on Symmetric Spaces

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

1. Introduction. The result to be proved in this article is that if u is a bounded harmonic function on a symmetric space X and x 0 any point in X then u has a limit along almost every geodesic in X starting at x 0 (Theorem 2.3). In the case when X is the unit disk with the non-Euclidean metric this result reduces to the classical Fatou theorem (for radial limits). When specialized to this case...

On Harmonic Functions on Trees

By a tree T we mean a connected graph such that every subgraph obtained from T by removing any of its edges is not connected. In what follows we will only consider trees in which we distinguish a vertex v0 as an origin. As usual we denote by V and E the set of vertices and the set of edges (respectively) of the tree. If v and w are the boundary vertices of an edge, we say that they are neighbou...

On a linear combination of classes of harmonic $p-$valent functions defined by certain modified operator

In this paper we obtain coefficient characterization‎, ‎extreme points and‎ ‎distortion bounds for the classes of harmonic $p-$valent functions‎ ‎defined by certain modified operator‎. ‎Some of our results improve‎ ‎and generalize previously known results‎.

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...