Characterization of eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers
Article Type
Research Article
Publication Title
Journal of Functional Analysis
Abstract
Let X be a rank one Riemannian symmetric space of noncompact type and Δ be the Laplace–Beltrami operator of X. The space X can be identified with the quotient space G/K where G is a connected noncompact semisimple Lie group of real rank one with finite center and K is a maximal compact subgroup of G. Thus G acts naturally on X by left translations. Through this identification, a function or measure on X is radial (i.e. depends only on the distance from eK), when it is invariant under the left-action of K. We consider right-convolution operators Θ on functions f on X defined by, Θ:f↦f⁎μ where μ is a radial (possibly complex) measure on X. These operators will be called multipliers. In particular Θ is a radial average when μ is a radial probability measure. Notable examples of radial averages are ball, sphere and annular averages and the heat operator. In this paper we address problems of the following type: Fix a multiplier, in particular an averaging operator Θ. Suppose that {fk}k∈Z is a bi-infinite sequence of functions on X such that for all k∈Z, Θfk=Afk+1 and ‖fk‖A∈C, M>0 and a suitable norm ‖⋅‖. From this hypothesis, we try to infer that f0, hence every fk, is an eigenfunction of Δ.
DOI
10.1016/j.jfa.2020.108737
Publication Date
12-15-2020
Recommended Citation
Naik, Muna and Sarkar, Rudra P., "Characterization of eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers" (2020). Journal Articles. 6.
https://digitalcommons.isical.ac.in/journal-articles/6