Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-13T07:56:42.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Variations on a theorem of Cowling and Price with applications to nilpotent Lie groups

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

S. Parui  and
S. Thangavelu
S. Parui
Affiliation:
Stat.-Math. Division Indian Statistical Institute203 B.T. Road Kolkata 700108India e-mail: sanjuparui@gmai.com
S. Thangavelu
Affiliation:
Department of Mathematics Indian Institute of ScienceBangalore 560012India e-mail: veluma@math.iisc.emet.in
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper we prove a new version of the Cowling-Price theorem for Fourier transforms on Rn. Using this we formulate and prove an uncertainty principle for operators. This leads to an analogue of the Cowling-Price theorem for nilpotent Lie groups. We also prove an exact analogue of the Cowling-Price theorem for the Heisenberg group.

Keywords

Weyl transformnilpotent Lie groupsheat kernelHermite and special Hermite functions.

MSC classification

Secondary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 11 - 27
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Astengo, F., Cowling, M., Di Blasio, B. and Sundari, M., ‘Hardy's uncertainty principle on certain Lie groups’, J. London Math. Soc. 62 (2000), 461–472.CrossRefGoogle Scholar
[2]Bagchi, S. C. and Ray, S. K., ‘Uncertainty principles like Hardy's theorem on some Lie groups’, J. Aust. Math. Soc. 65 (1999), 289–302.CrossRefGoogle Scholar
[3]Bakiouti, A. and Salah, N. B., ‘The version of Hardy's theorem on nilpotent Lie groups’, Forum Math. 18 (2006), 245–262.Google Scholar
[4]Bonami, A., Demange, B. and Jaming, P., ‘Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms’, Rev. Math. Iberoamericano 19 (2003), 23–55.Google Scholar
[5]Cowling, M. and Price, J., ‘Generalisations of Heisenberg's inequality’, in: Harmonic analysis (eds. Mauceri, G., Ricci, F. and Weiss, G.), Lecture Notes in Math. 992 (Springer, Berlin, 1983).Google Scholar
[6]Hardy, G. H., ‘A theorem concerning Fourier transforms’, J. London Math. Soc. 8 (1933), 227–231.CrossRefGoogle Scholar
[7]Kaniuth, E. and Kumar, A., ‘Hardy's theorem for simply connected nilpotent Lie groups’, Proc. Cambridge Philos. Soc. 131 (2001), 487–494.CrossRefGoogle Scholar
[8]Mauceri, G., ‘The Weyl transform and bounded operators on L p (Rn)’, J. Funct. Anal. 39 (1980), 408–429.CrossRefGoogle Scholar
[9]Pfannschmidt, C., ‘A generalization of the theorem of Hardy: A most general version of the uncertainty principle for Fourier integrals’, Math. Nachr. 182 (1996), 317–327.CrossRefGoogle Scholar
[10]Sitaram, A., Sundari, M. and Thangavelu, S., ‘Uncertainty principles on certain Lie groups’, Proc. Indian Acad. Sci. 105 (1995), 135–151.Google Scholar
[11]Thangavelu, S., Lectures on Hermite and Laguerre expansions, Math. Notes. 42 (Princeton University Press, Princeton, NJ, 1993).CrossRefGoogle Scholar
[12]Thangavelu, S., ‘On Paley-Wiener theorems for the Heisenberg group’, J. Funct. Anal. 115 (1993), 24–44.CrossRefGoogle Scholar
[13]Thangavelu, S., Harmonic analysis on the Heisenberg group, Progr. Math. 159 (Birkhäuser, Boston, 1998).CrossRefGoogle Scholar
[14]Thangavelu, S., ‘Hardy's theorem on the Heisenberg group revisited’, Math. Z. 242 (2002), 761–779.CrossRefGoogle Scholar
[15]Thangavelu, S., An introduction to the uncertainty principle: Hardy's theorem on Lie groups, Progr. Math. 217 (Birkhäuser, Boston, 2004).CrossRefGoogle Scholar