Relative Fatou theorem for {$\alpha$}-harmonic functions in Lipschitz domains

A Fatou-type Theorem for Harmonic Functions on Symmetric Spaces

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...

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...

A Prevalent Transversality Theorem for Lipschitz Functions

This paper provides a version of the transversality theorem for a class of Lipschitz functions of the form f : R × C → Rn where C is a convex subset of a normed vector space Z indexing the parameters in the problem. The set C may be infinite-dimensional, and the notion of generic used is the measure-theoretic notion of prevalence introduced by Hunt, Sauer and Yorke (1992) and Christensen (1974)...

(DELTA,GAMMA, 2)-BESSEL LIPSCHITZ FUNCTIONS IN THE SPACE L_{2,ALPHA}(R+)

Using a generalized translation operator, we obtain a generalization of Theorem 5 in [4] for the Bessel transform for functions satisfying the (delta;gamma ; 2)-BesselLipschitz condition in L_{2;alpha}(R+).