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The Fourier Transform on harmonic manifolds of purely exponential volume growth

Biswas, Kingshook; Knieper, Gerhard; Peyerimhoff, Norbert

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


Authors

Kingshook Biswas

Gerhard Knieper



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.

Citation

Biswas, K., Knieper, G., & Peyerimhoff, N. (2021). The Fourier Transform on harmonic manifolds of purely exponential volume growth. Journal of Geometric Analysis, 31(1), 126-163. https://doi.org/10.1007/s12220-019-00253-9

Journal Article Type Article
Acceptance Date Jul 29, 2019
Online Publication Date Aug 9, 2019
Publication Date 2021-01
Deposit Date Aug 8, 2019
Publicly Available Date Aug 9, 2020
Journal Journal of Geometric Analysis
Print ISSN 1050-6926
Electronic ISSN 1559-002X
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 31
Issue 1
Pages 126-163
DOI https://doi.org/10.1007/s12220-019-00253-9

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