On Pointwise converse of Fatou's theorem for Euclidean and Real hyperbolic spaces

TL;DR

This paper extends classical boundary behavior results of harmonic functions to more general settings, including hyperbolic spaces and solutions to the heat equation, revealing new asymptotic properties of eigenfunctions.

Contribution

It generalizes boundary behavior theorems for harmonic functions to hyperbolic spaces and nonnegative eigenfunctions, and explores their large-time asymptotics.

Findings

Boundary behavior results hold for more general approximate identities.

Eigenfunctions of the Laplace-Beltrami operator exhibit specific asymptotic behaviors.

Generalization of large time behavior of heat equation solutions on hyperbolic space.

Abstract

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 nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $\mathbb H^n$. We shall also prove a generalization of a result regarding large time behavior of solution of the heat equation proved in \cite{Re}. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $\mathbb H^n$.