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$L^p$ version of Hardy's theorem on semisimple Lie groups


Authors: E. K. Narayanan and S. K. Ray
Journal: Proc. Amer. Math. Soc. 130 (2002), 1859-1866
MSC (2000): Primary 22E30; Secondary 22E46, 43A30
DOI: https://doi.org/10.1090/S0002-9939-02-06272-X
Published electronically: January 16, 2002
MathSciNet review: 1887035
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Abstract: We prove an analogue of the $L^p$ version of Hardy's theorem on semisimple Lie groups. The theorem says that on a semisimple Lie group, a function and its Fourier transform cannot decay very rapidly on an average.


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  • 1. Anker J. P. The Spherical Fourier transform of Rapidly decreasing functions. A simple proof of a characterisation due to Harish-Chandra, Helgason, Trombi and Varadarajan, J. Funct. Anal. 96 (1991), 331-349. MR 92d:22008
  • 2. Astengo F. , Cowling M. G. , Di Blasio B. , Sundari M. Hardy's uncertainty principle on some Lie groups, J. London Math. Soc.(2) 62 (2000).no:2 461-472. CMP 2001:01
  • 3. Cowling M. G. , Price J. F. Generalizations of Heisenberg's Inequality, Harmonic Analysis (G. Mauceri, F. Ricci, G. Weiss, eds.), LNM, no. 992, Springer, Berlin, 1983, 443-449. MR 86g:42002b
  • 4. Bagchi S. C. , Ray S. Uncertainty principles like Hardy's theorem on some Lie groups, J. Austral. Math. Soc. (series A) 65(1998), 289-302. MR 99k:43001
  • 5. Cowling M. G. , Sitaram A. , Sundari M. Hardy's Uncertainty Principle on Semisimple Lie Groups, Pacific. J. Math. 192 (2000), 293-296. MR 2001c:22007
  • 6. Ebata M. , Eguchi M. , Koizumi S. , Kumahara K. A generalisation of the Hardy theorem to semisimple Lie groups, Proc. Japan. Acad. , Ser. A, Math. Sci. 75 (1999), 113-114. MR 2001c:22008
  • 7. Eguchi M. , Koizumi S. , Kumahara K. $L^p$ version of the Hardy theorem for motion groups, J. Austral. Math. Soc. (series A), 68 (2000), 55-67. MR 2001f:43005
  • 8. Folland G. B. , Sitaram A. The Uncertainty Principle: A Mathematical Survey, J. Fourier Anal. Appl. 3 (1997), 207-238. MR 98f:42006
  • 9. Gangolli R. , Varadarajan V. S. Harmonic Analysis of Spherical functions on Real Reductive Groups (Springer-Verlag, 1988). MR 89m:22015
  • 10. Hardy G. A theorem concerning Fourier transforms, J. London Math. Soc. 8 (1933), 227-231.
  • 11. Harish-Chandra On the theory of the Eisenstein integral, Harish-Chandra collected papers, Vol. 4, Narosa Publishing House, New Delhi, 1985.
  • 12. Helgason S. Groups and Geometric Analysis. Integral Geometry, Invariant Differential operators and Spherical functions, Academic Press, Orlando, 1984. MR 86c:22017
  • 13. Helgason S. Geometric Analysis on Symmetric spaces, AMS, Mathematical Surveys and Monographs, Vol 39, 1994. MR 96h:43009
  • 14. Hörmander L. A uniqueness theorem of Beurling for Fourier transform pairs, Arkiv för Matematik, 29 (2)(1991), 237-240. MR 93b:42016
  • 15. Knapp A. W. Representation Theory of Semisimple Lie Groups, an overview based on examples, Princeton Univ. Press, Princeton, NJ, 1986. MR 87j:22022
  • 16. Kaniuth E. , Kumar A. Hardy's theorem for simply connected nilpotent Lie groups, Preprint, 2000.
  • 17. Narayanan E. K., Ray S. Hardy's theorem on Symmetric spaces of noncompact type, Preprint, 2000.
  • 18. Ray S. Uncertainty principles on two step nilpotent Lie groups, to appear in Proc. Ind. Acad. Sci.
  • 19. Sarkar R. P. Revisiting Hardy's theorem on $SL(2, {\mathbb R}),$ Preprint, 2000.
  • 20. Sitaram A., Sundari M. An analogue of Hardy's theorem for very rapidly decreasing functions on semisimple Lie groups, Paciffic J. Math. 177 (1997), 187-200. MR 99a:22018
  • 21. Sengupta J. An analogue of Hardy's theorem on semisimple Lie groups Proc. AMS. 128 (2000), no:8, 2493-2499. MR 2000k:43004
  • 22. Terras A. Harmonic Analysis on Symmetric Spaces and Applications, Vols. 1 and 2, Springer-Verlag, 1988. MR 87f:22010; MR 89k:22017
  • 23. Warner G. , Harmonic analysis on Semisimple Lie groups, Vol. 2, Springer-Verlag, 1972. MR 58:16980

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Additional Information

E. K. Narayanan
Affiliation: Stat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560 059, India
Address at time of publication: Department of Mathematics & Computer Sciences, Bar-Ilan University, 52900 Ramat-Gan, Israel
Email: naru@isibang.ac.in, naru@macs.biu.ac.il

S. K. Ray
Affiliation: Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Calcutta 700035, India
Address at time of publication: Department of Mathematics, Indian Institute of Technology, Kanpur, U.P.-208016, India
Email: res9601@www.isical.ac.in, skray@iitk.ac.in

DOI: https://doi.org/10.1090/S0002-9939-02-06272-X
Keywords: Hardy's theorem, uncertainty principle, semisimple Lie groups
Received by editor(s): July 25, 2000
Received by editor(s) in revised form: January 2, 2001
Published electronically: January 16, 2002
Additional Notes: This research was supported by NBHM, India
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2002 American Mathematical Society

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