Radon Transform on a Harmonic Manifold
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Radon Transform on a Harmonic Manifold François Rouvière1 · Laboratoire J. A. Dieudonné1 Received: 11 June 2020 / Accepted: 7 October 2020 © Mathematica Josephina, Inc. 2020
Abstract We extend to a large class of noncompact harmonic manifolds the inversion formulas for the Radon transform on horospheres in hyperbolic spaces or Damek–Ricci spaces. Horospheres are defined here as level hypersurfaces of Busemann functions. The proof uses harmonic analysis on the manifolds considered, developed in a recent paper by Biswas, Knieper and Peyerimhoff; we also give a concise proof of their Fourier inversion theorem for harmonic manifolds. Keywords Harmonic manifold · Radon transform · Harmonic analysis · Busemann function · Horosphere Mathematics Subject Classification 44A12 · 43A32 · 53C25
Introduction Harmonic analysis and integral geometry have close links with each other. In various geometric frameworks the (generalized) Fourier transform of a function u is obtained by integrating it against eigenfunctions of the Laplace operator. Decomposing this integral according to the level sets of an eigenfunction, one can prove a so-called «projection slice theorem» which links the Radon transform Ru, given by the integrals of u over such sets, with the Fourier transform of u. A Fourier inversion formula for u may then lead to an inversion formula for the integral transform R. One of the simplest examples is given by the exponentials x → e2iπ ξ,x , eigenfunctions of the Laplace operator in Rn , whose level sets are the hyperplanes ξ, x = constant. The classical Fourier inversion theorem yields a Radon inversion formula which reconstructs u from its integrals over hyperplanes. Among many examples of this problem let us mention Helgason’s Radon transform over horospheres of a symmetric space of the noncompact type [12]. As symmetric
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François Rouvière [email protected] Université Côte d’Azur, Nice, France
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F. Rouvière, L. J. A. Dieudonné
spaces of rank one, the hyperbolic spaces are a special case, which also extends in a different direction to a large class of spaces (non symmetric in general) known as Damek–Ricci spaces or harmonic N A groups [9]. The purpose of this note is to extend the latter cases one step further, to all simply connected harmonic manifolds with purely exponential volume growth. For this we shall use harmonic analysis on these manifolds as developed in a recent paper by Biswas, Knieper and Peyerimhoff [4]. There are no Lie groups here; horospheres are defined by means of Busemann functions. In Sect. 1 we recall basic facts about harmonic manifolds, Busemann functions and we give explicit geometric expressions of these functions and the corresponding horospheres for hyperbolic spaces and for Damek–Ricci spaces. In Sect. 2 we summarize the main results of [4] in harmonic analysis, radial then non radial; for the reader’s convenience we provide a concise version of the proof of their Fourier inversion theorem (Theorem 7). In Sect. 3 we introduce the horosphere Radon transform R on a harmonic manif
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