Fatou Theorem and Its Converse for Positive Eigenfunctions of the Laplace–Beltrami Operator on Harmonic NA Groups
Article Type
Research Article
Publication Title
Journal of Geometric Analysis
Abstract
We prove a Fatou-type theorem and its converse for certain positive eigenfunctions of the Laplace–Beltrami operator L on a Harmonic NA group. We show that a positive eigenfunction u of L with eigenvalue β2- ρ2 , where β∈ (0 , ∞) , has admissible limit in the sense of Korányi, precisely at those boundary points where the strong derivative of the boundary measure of u exists. Moreover, the admissible limit and the strong derivative are the same. This extends a result of Ramey and Ullrich regarding nontangential convergence of positive harmonic functions on the Euclidean upper half space.
DOI
https://10.1007/s12220-023-01279-w
Publication Date
7-1-2023
Recommended Citation
Ray, Swagato K. and Sarkar, Jayanta, "Fatou Theorem and Its Converse for Positive Eigenfunctions of the Laplace–Beltrami Operator on Harmonic NA Groups" (2023). Journal Articles. 3662.
https://digitalcommons.isical.ac.in/journal-articles/3662