The Fourier Transform on harmonic manifolds of purely exponential volume growth

Biswas, Kingshook; Knieper, Gerhard; Peyerimhoff, Norbert

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
• Public URL: https://durham-repository.worktribe.com/output/1290636