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

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 $\lambda \in \mathbb{R}\setminus \{0\}$.

References in corpus (1)

• Characterization of almost $L^p$-eigenfunctions of the Laplace-Beltrami operator

Cited by in corpus (2)

• Analysis of $L^p$-type estimates of Poisson transform on Homogeneous Trees
• Function with its Fourier transform supported on annulus and eigenfunction of Laplacian