Bulletin of the Australian Mathematical Society

Research Article

The Helgason Fourier transform on a class of nonsymmetric harmonic spaces

Francesca Astengoa1, Roberto Camporesia1 and Bianca Di Blasioa1

a1 Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129, Torino Italy

Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.

(Received June 11 1996)