Eigenfunction of the Laplacian as a degenerate case of a function with its Fourier transform supported in an annulus
type: Item
instance of: scholarly article
title: Eigenfunction of the Laplacian as a degenerate case of a function with its Fourier transform supported in an annulus
zbMATH Open document ID: 1397.43006
DOI: 10.4171/PRIMS/54-2-4
author: Rudra P. Sarkar
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2018-05-08T00:00:00Z
review text: The Paley-Wiener Theorem gives a characterization of functions whose Fourier transforms vanish outside finite intervals symmetric around the origin. This result has been extended to higher dimensions and abstract spaces. An inverse Paley-Wiener theorem gives necessary and sufficient conditions that a function must satisfy in order for its Fourier transform to be compactly supported in a ball around the origin. This type of inverse theorems involves holomorphic extension of the function and some growth conditions on the extension. Another type of inverse Paley-Wiener theorems does not use complex analysis and holomorphic extension, but uses real variable techniques. Some inverse theorems give criteria involving norm estimates on integral powers of differential operators, such as the Laplacian, acting on the function. The aim of the paper under review is to give a characterization of the eigenfunctions of the Laplace-Beltrami operator as a degenerate case of an inverse Paley-Wiener theorem on rank-1 Riemannian symmetric spaces of noncompact type. To give the reader a glimpse of some of the results presented in the paper, let \(X\) be a rank-1 Riemannian symmetric space of noncompact type, \(\Delta\) be the Laplace-Beltrami Operator on \(X\) and \(B\) be its maximal distinguished boundary. Let \(L^{2,\infty}(X)\) be the Lorentz space of weak-\(L^2\) functions on \(X,\) and \(-\rho^2\) be the bottom of the \(L^2\) spectrum of \(\Delta.\) Suppose that \(f\in L^{2,\infty}(X)\) and there exist \(c_1\geq \rho^2\) and \(0<c_2 \leq 1/\rho^2\) such that \[ \lim_{n\rightarrow \infty}\left\| \Delta^n f\right\|_{2, \infty}^{1/n} =c_1, \quad \lim_{n\rightarrow \infty}\left\| \Delta^{-n} f\right\|_{2, \infty}^{1/n} =c_2. \] It is shown that {\parindent=0.7cm\begin{itemize}\item[--] \(c_1 c_2 \geq 1\). \item[--] If \(c_1c_2>1,\) then the Fourier transform \(\hat{f}\) of \(f\) is supported on the annulus, \({\mathcal A}_\beta^\alpha =[\beta, \alpha]\times B,\) around the origin, wher…
reviewed by: Ahmed I. Zayed
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 6866869
zbMATH Keywords: eigenfunctions of Laplacian
zbMATH Keywords: Riemannian symmetric spaces
zbMATH Keywords: Lorentz spaces
zbMATH Keywords: inverse Paley-Wiener theorem
MaRDI profile type: Publication
Wikidata QID: Q125875409
full work available at URL: https://doi.org/10.4171/prims/54-2-4
OpenAlex ID: W2796452639
description: scientific article; zbMATH DE number 6866869