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arXiv · 2012.01824

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

Jayanta Sarkar
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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 $\mathbb H^n$.

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Jayanta Sarkar. 2020-12-03. On Pointwise converse of Fatou's theorem for Euclidean and Real hyperbolic spaces. https://doi.org/10.1007/s11856-022-2336-0

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