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The Fourier Transform on harmonic manifolds of purely exponential volume growth.

Biswas , Kingshook and Knieper, Gerhard and Peyerimhoff, Norbert (2021) 'The Fourier Transform on harmonic manifolds of purely exponential volume growth.', The journal of geometric analysis., 31 (1). pp. 126-163.

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)|−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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s12220-019-00253-9
Publisher statement:This is a post-peer-review, pre-copyedit version of an article published in The journal of geometric analysis. The final authenticated version is available online at: https://doi.org/10.1007/s12220-019-00253-9
Date accepted:29 July 2019
Date deposited:08 August 2019
Date of first online publication:09 August 2019
Date first made open access:09 August 2020

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