"Fatou Theorem and Its Converse for Positive Eigenfunctions of the Lapl" by Swagato K. Ray and Jayanta Sarkar
 

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

Share

COinS