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

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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 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.

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Acknowledgements

The author would like to thank Swagato K. Ray for suggesting this problem and for many useful discussions during the course of this work.

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Authors and Affiliations

  1. Stat-Math Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata, 700108, India

    Jayanta Sarkar

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  1. Jayanta Sarkar
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Correspondence to Jayanta Sarkar.

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The author is supported by a research fellowship from the Indian Statistical Institute.

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Sarkar, J. On the pointwise converse of Fatou’s theorem for Euclidean and real hyperbolic spaces. Isr. J. Math. 250, 179–209 (2022). https://doi.org/10.1007/s11856-022-2336-0

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  • DOI: https://doi.org/10.1007/s11856-022-2336-0

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