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Uncertainty principles on two step nilpotent Lie groups

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Abstract

We extend an uncertainty principle due to Cowling and Price to two step nilpotent Lie groups, which generalizes a classical theorem of Hardy. We also prove an analogue of Heisenberg inequality on two step nilpotent Lie groups.

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References

  1. Astengo F, Cowling M G, Di Blasio B and Sundari M, Hardy’s uncertainty principle on some Lie groups,J. London Math. Soc. (to appear)

  2. Benson C, Jenkins J and Ratcliff G, On Gelfand pairs associated to solvable Lie groups,Trans. Am. Math. Soc. 321 (1990) 85–116

    Article  MATH  MathSciNet  Google Scholar 

  3. Bagchi S C and Ray S, Uncertainty principles like Hardy’s theorem on some Lie groups,J. Austral. Math. Soc. A65 (1998) 289–302

    Article  MathSciNet  Google Scholar 

  4. Chandrasekharan K, Classical Fourier transforms (Springer Verlag) (1989)

  5. Cowling M G, Sitaram A and Sundari M, Hardy’s uncertainty principle on semisimple Lie groups,Pacific J. Math. 192 (2000) 293–296

    MATH  MathSciNet  Google Scholar 

  6. Cowling M G and Price J F, Generalizations of Heisenberg’s inequality, in: Harmonic analysis (eds) G Mauceri, F Ricci and G Weiss (1983) LNM, no. 992 (Berlin: Springer) 443–449

    Chapter  Google Scholar 

  7. Corwin L J and Greenleaf F P, Representations of nilpotent Lie groups and their applications, Part 1 — Basic theory and examples (NY: Cambridge Univ. Press, Cambridge) (1990)

    MATH  Google Scholar 

  8. Ebata M, Eguchi M, Koizumi S and Kumahara K, A generalisation of the Hardy theorem to semisimple Lie groups,Proc. Japan Acad. Math. Sci. A75 (1999) 113–114

    MathSciNet  Google Scholar 

  9. Ebata M, Eguchi M, Koizumi S and Kumahara K,L p version of the Hardy theorem for motion groups,J. Austral. Math. Soc. A68 (2000) 55–67

    Google Scholar 

  10. Folland G B, A course in abstract harmonic analysis, (London: CRC Press) (1995)

    MATH  Google Scholar 

  11. Folland G B, Lectures on partial differential equations (New Delhi: Narosa Pub. House) (1983)

    MATH  Google Scholar 

  12. Folland G B and Sitaram A, The uncertainty principles: A mathematical survey,J. Fourier Anal. Appl. 3 (1997) 207–238

    Article  MATH  MathSciNet  Google Scholar 

  13. Garofalo N and Lanconeli E, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation,Ann. Inst. Fourier 40 (1990) 313–356

    MATH  Google Scholar 

  14. Hörmander L, A uniqueness theorem of Beurling for Fourier transform pairs,Ark. Math. 29 (1991) 237–240

    Article  MATH  Google Scholar 

  15. Jacobson N, Basic Algebra, (New Delhi: Hindustan Publishing Co.) (1993) vol. 1

    Google Scholar 

  16. Kaniuth E and Kumar A, Hardy’s theorem for simply connected nilpotent Lie groups,Proc. Cambridge Philos. Soc. (to appear)

  17. Lipsman R L and Rosenberg J, The behavior of Fourier transform for nilpotent Lie groups,Trans. Am. Math. Soc. 348 (1996) 1031–1050

    Article  MATH  MathSciNet  Google Scholar 

  18. Moore C C and Wolf J A, Square integrable representations of nilpotent Lie groups,Trans. Am. Math. Soc. 185 (1973) 445–462

    Article  MathSciNet  Google Scholar 

  19. Muller D and Ricci F, Solvability for a class of doubly characteristic differential operators on two step nilpotent Lie groups,Ann. Math. 143 (1996) 1–49

    Article  MathSciNet  Google Scholar 

  20. Narayanan E K and Ray S K,Lp version of Hardy’s theorem on semisimple Lie groups,Proc. Am. Math. Soc. (to appear)

  21. Park R, A Paley-Wiener theorem for all two-and three-step nilpotent Lie groups,J. Funct. Anal. 133 (1995) 277–300

    Article  MATH  MathSciNet  Google Scholar 

  22. Pati V, Sitaram A, Sundari M and Thangavelu S, An uncertainty principle for eigenfunction expansions,J. Fourier Anal. Appl. 5 (1996) 427–433

    MathSciNet  Google Scholar 

  23. Ray S K, Uncertainty principles on some Lie groups, Ph.D. Thesis (Indian Statistical Institute) (1999)

  24. Sundari M, Hardy’s theorem for the n-dimensional Euclidean motion group,Proc. Am. Math. Soc. 126 (1998) 1199–1204

    Article  MATH  MathSciNet  Google Scholar 

  25. Sengupta J, An analogue of Hardy’s theorem for semi-simple Lie groups,Proc. Am. Math. Soc. 128 (2000) 2493–2499

    Article  MATH  Google Scholar 

  26. Strichartz R F,L p harmonic analysis and radon transforms on the Heisenberg groups,J Funct. Anal. 96 (1991) 350–406

    Article  MATH  MathSciNet  Google Scholar 

  27. Sitaram A and Sundari M, An analogue of Hardy’s theorem for very rapidly decreasing functions on semisimple Lie groups,Pacific J. Math. 177 (1997) 187–200

    Article  MATH  MathSciNet  Google Scholar 

  28. Sitaram A, Sundari M and Thangavelu S, Uncertainty principles on certain Lie groups,Proc. Ind. Acad. Sci. (Math. Sci.) 105 (1995) 135–151

    MATH  MathSciNet  Google Scholar 

  29. Thangavelu S, Some uncertainty inequalities,Proc. Ind. Acad. Sci. (Math. Sci.) 100 (1990) 137–145

    MATH  MathSciNet  Google Scholar 

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  1. Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, 700 035, Kolkata, India

    S. K. Ray

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  1. S. K. Ray
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Ray, S.K. Uncertainty principles on two step nilpotent Lie groups. Proc. Indian Acad. Sci. (Math. Sci.) 111, 293–318 (2001). https://doi.org/10.1007/BF02829598

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