Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-19T20:34:40.536Z Has data issue: false hasContentIssue false


Part of: Lie groups

Published online by Cambridge University Press:  19 January 2010

Department of Mathematics, Faculty of Sciences at Sfax, Route de Soukra, 3038, Sfax, Tunisia (email:
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India (email:
For correspondence; e-mail:
Rights & Permissions [Opens in a new window]


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.

We formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

MSC classification

Research Article
Copyright © Australian Mathematical Publishing Association Inc. 2010


The first author was supported by D.G.R.S.R.T., Research Unity 00 UR 1501 and the second author by a J.C. Bose Fellowship from DST.


[1]Baklouti, A. and Kaniuth, E., ‘On Hardy’s uncertainty principle for connected nilpotent Lie groups’, Math. Z. 259(2) (2008), 233247.Google Scholar
[2]Baklouti, A. and Kaniuth, E., ‘On Hardy’s uncertainty principle for solvable locally compact groups’, J. Fourier Anal. Appl., to appear.Google Scholar
[3]Baklouti, A. and Ben Salah, N., ‘The Lp-Lq version of Hardy’s theorem on nilpotent Lie groups’, Forum Math. 18(2) (2006), 245262.CrossRefGoogle Scholar
[4]Baklouti, A. and Ben Salah, N., ‘On theorems of Beurling and Cowling–Price for certain nilpotent Lie groups’, Bull. Sci. Math. 132(6) (2008), 529550.Google Scholar
[5]Baklouti, A., Smaoui, K. and Ludwig, J., ‘Estimate of L p-Fourier transform norm on nilpotent Lie groups’, J. Funct. Anal. 199 (2003), 508520.Google Scholar
[6]Cowling, M. and Price, J., ‘Generalizations of Heisenberg’s inequality’, in: Harmonic Analysis, Lecture Notes in Mathematics, 992 (eds. Mauceri, G., Ricci, F. and Weiss, G.) (Springer, Berlin, 1983).Google Scholar
[7]Folland, G. and Sitaram, A., ‘The uncertainty principle: a mathematical survey’, J. Fourier Anal. Appl. 3(3) (1997), 207238.Google Scholar
[8]Hardy, G. H., ‘A theorem concerning Fourier transforms’, J. London Math. Soc. 8 (1933), 227231.Google Scholar
[9]Kaniuth, E. and Kumar, A., ‘Hardy’s theorem for nilpotent Lie groups’, Math. Proc. Cambridge Philos. Soc. 131 (2001), 487494.CrossRefGoogle Scholar
[10]Miyachi, A., ‘A generalization of a theorem of Hardy’, Harmonic Analysis Seminar held at Izunagaoka, Shizuoka-ken, Japan, (1997), 44–51.Google Scholar
[11]Parui, S. and Thangavelu, S., ‘Variations on a theorem of Cowling and Price with applications to nilpotent Lie groups’, J. Aust. Math. Soc. 82 (2007), 1127.CrossRefGoogle Scholar
[12]Thangavelu, S., ‘Hardy’s theorem on the Heisenberg group revisited’, Math. Z. 242 (2002), 761779.Google Scholar
[13]Thangavelu, S., An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups, Progress in Mathematics, 217 (Birkhäuser, Boston, MA, 2004).CrossRefGoogle Scholar
[14]Thangavelu, S., ‘A survey of Hardy type theorems’, in: Advances in Analysis, (eds. Begehr, H. G. W., Gilbert, R. P., Muldoon, M. E. and Wong, M. W.) (World Scientific, Singapore, 2005), pp. 3970.Google Scholar