Fourier restriction theorem and characterization of weak L2 eigenfunctions of the Laplace-Beltrami operator

In this paper we prove that for λ∈R∖{0} the Poisson transform PλF of F∈L2(K/M) belongs to L2,∞(G/K). This complements an earlier result of Lohoué and Rychener. Using this we then prove an analogue of Tomas-Stein restriction theorem for Riemannian symmetric spaces of noncompact type with real rank one. By using this estimate of the Poisson transform we also characterize all weak L2 eigenfunction of the Laplace-Beltrami operator of Riemannian symmetric spaces of noncompact type with real rank one and eigenvalue −(λ2+ρ2) for λ∈R∖{0}.