On the pointwise converse of Fatou’s theorem for Euclidean and real hyperbolic spaces

In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space ℝ+n+1. We show that similar results remain valid for more general approximate identities. We apply this result to prove a result regarding boundary behavior of certain nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn. We shall also prove a generalization of a result regarding large time behavior of a solution of the heat equation proved in [17]. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn.