Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T02:52:57.796Z Has data issue: false hasContentIssue false
  • English
  • Français

The Helgason Fourier transform on a class of nonsymmetric harmonic spaces

Published online by Cambridge University Press:  17 April 2009

Francesca Astengo ,
Roberto Camporesi  and
Bianca Di Blasio
Francesca Astengo
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129, TorinoItaly
Roberto Camporesi
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129, TorinoItaly
Bianca Di Blasio
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129, TorinoItaly
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 3 , June 1997 , pp. 405 - 424
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Anker, J.-Ph., Damek, E., and Yacoub, C., ‘Spherical analysis on harmonic AN groups’, (preprint).Google Scholar
[2]Cowling, M., Dooley, A.H., Korányi, A., and Ricci, F., ‘An approach to symmetric spaces of rank one via groups of Heisenberg type’, J. Geom. Anal. (to appear).Google Scholar
[3]Damek, E., ‘A Poisson kernel on Heisenberg type nilpotent groups’, Colloq. Math. 53 (1987), 239–247.CrossRefGoogle Scholar
[4]Damek, E. and Ricci, F., ‘Harmonic analysis on solvable extensions of H-type groups’, J. Geom. Anal. 2 (1992), 213–248.CrossRefGoogle Scholar
[5]Damek, E. and Ricci, F., ‘A class of nonsymmetric harmonic Riemannian spaces’, Bull. Amer. Math. Soc. 27 (1992), 139–142.CrossRefGoogle Scholar
[6]Di Blasio, B., ‘Paley-Wiener type theorems on harmonic extension of H-type groups’, Monatsh. Math. 123 (1997), 21–42.CrossRefGoogle Scholar
[7]Di Blasio, B., ‘An extension of the theory of Gelfand pairs to radial functions on Lie groups’, (preprint), Boll. Un. Mat. Ital. (to appear).Google Scholar
[8]Helgason, S., Geometric Analysis on Symmetric Spaces, Math. Surveys and Monographs 39 (American Mathematical Society, Providence RI, 1994).Google Scholar
[9]Kaplan, A., ‘Fundamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms’, Trans. Amer. Math. Soc. 258 (1980), 147–153.CrossRefGoogle Scholar
[10]Korányi, A., ‘Geometric properties of Heisenberg type groups’, Adv. Math. 56 (1985), 28–38.CrossRefGoogle Scholar
[11]Ricci, F., ‘The spherical transform on harmonic extensions of H-type groups’ Rend. Sem. Mat. Univ. Pol. Torino 50 (1992), 381–392.Google Scholar