Characterization of eigenfunctions of the Laplace–Beltrami operator using Fourier multipliers

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‖<M for some constants 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.