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

TL;DR

This paper extends the Fourier restriction theorem to rank-one noncompact Riemannian symmetric spaces and characterizes weak L^2 eigenfunctions of the Laplace--Beltrami operator using Poisson transform estimates.

Contribution

It proves the Fourier restriction theorem for p=2 on rank-one noncompact symmetric spaces and characterizes weak L^2 eigenfunctions via Poisson transform estimates.

Findings

Established Fourier restriction theorem for p=2 on these spaces.

Characterized weak L^2 eigenfunctions of the Laplace--Beltrami operator.

Extended previous results to a broader class of symmetric spaces.

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