The Fourier transform on harmonic manifolds

Abstract: There is a well-known theory of a Fourier transform on the n-dimensional hyperbolic space \(H^n\) which is analogous to that of the classical Fourier transform on Euclidean space \(R^n\). More generally, this Fourier transform can be defined for symmetric spaces G/K, and also for certain solvable Lie groups called "harmonic NA groups", or "Damek-Ricci spaces". We generalize these cases to the class of noncompact Riemannian manifolds known as "harmonic manifolds". We prove a Fourier inversion formula and a Plancherel theorem. These results were first proved for harmonic manifolds of negative curvature by the speaker, and then extended to the class of Gromov hyperbolic harmonic manifolds in joint work with G. Knieper and N. Peyerimhoff.

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