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

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 ${\beta }^2-{\rho }^2$, $\beta \in (0,\mathrm{\infty })$, 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.