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arXiv · 1204.1430

Fourier restriction Theorem and characterization of weak $L^2$ eigenfunctions of the Laplace--Beltrami operator

Pratyoosh Kumar
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Abstract

In this paper we prove the Fourier restriction theorem for $p=2$ on Riemannian symmetric spaces of noncompact type with real rank one which extends the earlier result proved in \cite[Theorem 1.1]{KRS}. This result depends on the weak $L^2$ estimates of the Poisson transform of $L^2$ function. By using this estimate of the Poisson transform we also characterizes all weak $L^2$ eigenfunction of the Laplace--Beltrami operator of Riemannian symmetric spaces of noncompact type with real rank one and eigenvalue $-(\lambda^2+\rho^2)$ for $\lambda\in\R\setminus\{0\}$.

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Pratyoosh Kumar. 2012-04-06. Fourier restriction Theorem and characterization of weak $L^2$ eigenfunctions of the Laplace--Beltrami operator. https://doi.org/10.1016/j.jfa.2013.10.009

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