The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

Article Type

Research Article

Publication Title

Journal of Geometric Analysis


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 non-compact 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)|-2dλfor f∈Cc∞(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 C> 0 is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon.

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Open Access, Green

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