Fatou theorem and its converse for positive eigenfunctions of the Laplace-Beltrami operator on Harmonic $NA$ groups

TL;DR

This paper establishes a Fatou theorem and its converse for positive eigenfunctions of the Laplace-Beltrami operator on Harmonic $NA$ groups, linking boundary limits with measure derivatives, extending Euclidean results.

Contribution

It extends Fatou-type theorems to Harmonic $NA$ groups, connecting boundary behavior of eigenfunctions with measure derivatives, a novel generalization beyond Euclidean spaces.

Findings

Positive eigenfunctions have admissible limits where the boundary measure's strong derivative exists.

Admissible limits coincide with the strong derivative of the boundary measure.

The results generalize Euclidean nontangential convergence theorems to Harmonic $NA$ groups.

Abstract

We prove a Fatou-type theorem and its converse for certain positive eigenfunctions of the Laplace-Beltrami operator $\mathcal{L}$ on a Harmonic $NA$ group. We show that a positive eigenfunction $u$ of $\mathcal{L}$ with eigenvalue $\beta^2-\rho^2$, $\beta\in (0,\infty)$, has an admissible limit in the sense of Kor\'anyi, 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.