The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

Abstract

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by h>0 the mean curvature of horospheres in X, and set ρ=h/2. Fixing a basepoint o∈X, for ξ∈∂X, denote by Bξ the Busemann function at ξ such that Bξ(o)=0. then for λ∈C the function e(iλ−ρ)Bξ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue −(λ2+ρ2). For a function f on X, we define the Fourier transform of f by f~(λ,ξ):=∫Xf(x)e(−iλ−ρ)Bξ(x)dvol(x) for all λ∈C,ξ∈∂X for which the integral converges. We prove a Fourier inversion formula f(x)=C0∫∞0∫∂Xf~(λ,ξ)e(iλ−ρ)Bξ(x)dλo(ξ)|c(λ)|−2dλ for f∈C∞c(X), where c is a certain function on R−{0}, λo is the visibility measure on ∂X with respect to the basepoint o∈X and C0>0 is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.