We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.

Durham Research Online
You are in:

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.


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
Download PDF
Publisher Web site:
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:
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

Save or Share this output

Look up in GoogleScholar