Eigenfunction of the Laplacian as a degenerate case of a function with its Fourier transform supported in an annulus
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, where \[ \alpha=\sqrt{ c_1 -\rho^2}, \quad \text{ and } \beta=\sqrt{ 1/c_2 -\rho^2}\, . \] \item[--] If \(c_1 c_2 =1,\) then \(f\) is an eigenfunction of \(\Delta\) with eigenvalue \(-c_1. \) \end{itemize}} Other variations of this result are obtained.
scientific article; zbMATH DE number 6866869