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An Analogue of Beurling's Theorem for the Heisenberg Group

Published online by Cambridge University Press:  17 April 2009

Jizheng Huang
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China, e-mail: hjzheng@pku.edu.cn; hpliu@pku.edu.cn
Heping Liu
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China, e-mail: hjzheng@pku.edu.cn; hpliu@pku.edu.cn
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In this paper, we prove an analogue of Beurling's theorem on the Heisenberg group. Then we derive some other versions of the uncertainty principle.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Astengo, F., Cowling, M., Di Blasio, B. and Sundari, M., ‘Hardy's uncertainty pronciple on some Lie groups’, J. London Math. Soc. 62 (2000), 461472.CrossRefGoogle Scholar
[2]Baclouti, A. and Ben, S. N., ‘The LP - Lq version of Hardy's theorem on nilpotent Lie groups’, Forum Math. 18 (2006), 245262.Google Scholar
[3]Bagchi, S.C. and Ray, S.K., ‘Uncertainty principles like Hardy's theorem on some Lie groups’, J. Aust. Math. Soc. Ser. A 65 (1999), 289302.CrossRefGoogle Scholar
[4]Bonami, A., Demange, B. and Jaming, P., ‘Hermite functions and uncertainty principles for the Fourier and the widowed Fourier transform’, Rev. Mat. Iberoamericana 19 (2003), 2355.CrossRefGoogle Scholar
[5]Folland, G.B., Harmoic analysis in phase space, Ann. Math. Stud. 122 (Princeton University Press, Princeton, NJ, 1989).Google Scholar
[6]Folland, G.B. and Sitaram, A., ‘The uncertainty principle: a mathematical survey’, J. Fourier Anal. Appl. 3 (1997), 207238.CrossRefGoogle Scholar
[7]Hömander, L., ‘A uniqueness theorem of Beurling for Fourier transform pairs’, Ark. Mat. 29 (1991), 237240.CrossRefGoogle Scholar
[8]Kaniuth, E. and Kumar, A., ‘Hardy's theorem for simply connected nilpotent Lie groups’, Math. Proc. Cambridge Philos. Soc. 131 (2001), 487494.CrossRefGoogle Scholar
[9]Kumar, A. and Bhatta, C.R., ‘An uncertainty principle like Hardy's theorem for nilpotent Lie groups’, J. Aust. Math. Soc. 77 (2004), 4753.CrossRefGoogle Scholar
[10]Ray, S.K., Uncertainty principles on two step nilpotent Lie groups, Proc. Ind. Acad. Sci. Math. Soc. 111, pp. 293318.CrossRefGoogle Scholar
[11]Sarkar, R.P. and Sengupta, J., ‘Beurling's theorem and characterization of heat kernel for Riemannian symmtric spaces of noncompact type’, arXiv: math. FA/0502514, V1, (2005).Google Scholar
[12]Sarkar, R.P. and Thangavelu, S., ‘On theorems of Beurling and Hardy for the Euclidean Motion group’, Tohoku. Math. J. 57 (2005), 335351.CrossRefGoogle Scholar
[13]Sitaram, A., Sundari, M. and Thangavelu, S., ‘Uncertainty principles on certain Lie groups’, Proc. Indian Acad. Sci. Math. Sci. 105 (1995), 135151.CrossRefGoogle Scholar
[14]Thangavelu, S., Lectures on Hermite and Laguerre expansions, Math. Notes 42 (Princeton University Press, Princeton, NJ, 1993).CrossRefGoogle Scholar
[15]Tangavelu, S., Harmonic analysis on the Heisenberg group, Progr. Math. 159 (Birkhäuser, Boston, 1998).CrossRefGoogle Scholar
[16]Thangavelu, S., ‘An analogue of Hardy's theorem for the Heisenberg group’, Colloq. Math. 87 (2001), 137145.CrossRefGoogle Scholar
[17]Thangavelu, S., ‘Revisiting Hardy's theorem for the Heisenberg group’, Math. Z. 242 (2002), 761779.CrossRefGoogle Scholar
[18]Thangavelu, S., An introduction to the uncertainty principle, Progr. Math. 217 (Birkhäuser, Boston, 2003).Google Scholar
[19]Thangavelu, S., ‘On theorems of Hardy, Gelfand-Shilov and Beurling for semisimple Lie groups’, Publ. Res. Inst. Math. Sci. 40 (2004), 311344.CrossRefGoogle Scholar