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Journal of the London Mathematical Society 2000 62(2):461-472; doi:10.1112/S0024610700001186
© 2000 by London Mathematical Society
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© The London Mathematical Society

Hardy's Uncertainty Principle on Certain Lie Groups

F. Astengo, M. Cowling, B. Di Blasio and M. Sundari

Dipartimento di Matematica, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy, astengo{at}calvino.polito.it
School of Mathematics, University of New South Wales Sydney NSW 2052, Australia, m.cowling{at}unsw.edu.au
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’ via della Ricerca Scientifica 1, 00133 Roma, Italy, diblasio{at}axp.mat.uniroma2.it
PO Box 5978, Jeddah 21432, Saudi Arabia, smadhava{at}memrbksa.com

Received 20 June 1998. Revision received 11 May 1999.

A theorem due to Hardy states that, if f is a function on R such that |f(x)| ≤ C e–{alpha}|x|2 for all x in R and Formula for all {xi} in R, where {alpha} > 0, ß > 0, and {alpha}ß > 1/4, then f = 0. A version of this celebrated theorem is proved for two classes of Lie groups: two-step nilpotent Lie groups and harmonic NA groups, the latter being a generalisation of noncompact rank-1 symmetric spaces. In the first case the group Fourier transformation is considered; in the second case an analogue of the Helgason–Fourier transformation for symmetric spaces is considered.


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