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

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