"The Fourier Transform on Harmonic Manifolds of Purely Exponential Volu" by Kingshook Biswas, Gerhard Knieper et al.
 

The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

Article Type

Research Article

Publication Title

Journal of Geometric Analysis

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 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.

First Page

126

Last Page

163

DOI

10.1007/s12220-019-00253-9

Publication Date

1-1-2021

Comments

Open Access, Green

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