Uncertainty principles of Ingham and Paley–Wiener on semisimple Lie groups

Article
  • 2 Downloads

Abstract

Classical results due to Ingham and Paley–Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on Riemmanian symmetric spaces of noncompact type.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Ben Saïd, S. Thangavelu and V. N. Dogga, Uniqueness of solutions to Schrödinger equations on H-type groups, Journal of the Australian Mathematical Society 95 (2013), 297–314.Google Scholar
  2. [2]
    M. Bhowmik, S. K. Ray and S. Sen, Around uncertainty principles of ingham-type on Rn, Tn and two step nilpotent lie groups, preprint, arXiv:1605.09616.Google Scholar
  3. [3]
    M. Bhowmik and S. Sen, An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group, J. Fourier Anal. Appl. 23 (2017), 1445–1464.Google Scholar
  4. [4]
    S. Bochner, Quasi-analytic functions, Laplace operator, positive kernels, Annals of Mathematics 51 (1950), 68–91.Google Scholar
  5. [5]
    S. Bochner and A. E. Taylor, Some theorems on quasi-analyticity for functions of several variables, American Journal of Mathematics 61 (1939), 303–329.Google Scholar
  6. [6]
    R. B. Burckel, An Introduction to Classical Complex Analysis. Vol. 1, Pure and Applied Mathematics, Vol. 82, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1979.Google Scholar
  7. [7]
    S. Chanillo, Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups, Indian Academy of Sciences. Proceedings. Mathematical Sciences 117 (2007), 325–331.Google Scholar
  8. [8]
    J. B. Conway, Functions of One Complex Cariable. II, Graduate Texts in Mathematics, Vol. 159, Springer-Verlag, New York, 1995.Google Scholar
  9. [9]
    M. Cowling and A. Nevo, Uniform estimates for spherical functions on complex semisimple Lie groups, Geometric and Functional Analysis 11 (2001), 900–932.Google Scholar
  10. [10]
    M. Cowling, A. Sitaram and M. Sundari, Hardy’s uncertainty principle on semisimple groups, Pacific Journal of Mathematics 192 (2000), 293–296.Google Scholar
  11. [11]
    M. Eguchi, Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces, Journal of Functional Analysis 34 (1979), 167–216.Google Scholar
  12. [12]
    L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Communications in Partial Differential Equations 31 (2006), 1811–1823.Google Scholar
  13. [13]
    L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, Uniqueness properties of solutions to Schrödinger equations, Bulletin of the American Mathematical Society 49 (2012), 415–442.Google Scholar
  14. [14]
    G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, Journal of Fourier Analysis and Applications 3 (1997), 207–238.Google Scholar
  15. [15]
    R. Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 101, Springer-Verlag, Berlin, 1988.Google Scholar
  16. [16]
    V. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 28, Springer-Verlag, Berlin, 1994.Google Scholar
  17. [17]
    S. Helgason, Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, Vol. 39, American Mathematical Society, Providence, RI, 1994.Google Scholar
  18. [18]
    S. Helgason, Groups and Geometric Analysis, Mathematical Surveys and Monographs, Vol. 83, American Mathematical Society, Providence, RI, 2000.Google Scholar
  19. [19]
    S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, American Mathematical Society, Providence, RI, 2001.Google Scholar
  20. [20]
    I. I. Hirschman, Jr., On the behaviour of Fourier transforms at infinity and on quasianalytic classes of functions, American Journal of Mathematics 72 (1950), 200–213.Google Scholar
  21. [21]
    A. E. Ingham, A note on Fourier transforms, Journal of the London Mathematical Society 9 (1934), 29–32.Google Scholar
  22. [22]
    A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Mathematical Series, Vol. 36, Princeton University Press, Princeton, NJ, 1986.Google Scholar
  23. [23]
    P. Koosis, The Logarithmic Integral. I, Cambridge Studies in Advanced Mathematics, Vol. 12, Cambridge University Press, Cambridge, 1998.Google Scholar
  24. [24]
    N. Levinson, On a class of non-vanishing functions, Proceedings of the London Mathematical Society 41 (1936), 393–407.Google Scholar
  25. [25]
    N. Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940.Google Scholar
  26. [26]
    J. Ludwig and D. Müller, Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups, Proceedings of the American Mathematical Society 142 (2014), 2101–2118.Google Scholar
  27. [27]
    R. E. A. C. Paley and N. Wiener, Notes on the theory and application of Fourier transforms. I, II, Transactions of the American Mathematical Society 35 (1933), 348–355.Google Scholar
  28. [28]
    R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, Vol. 19, American Mathematical Society, Providence, RI, 1987.Google Scholar
  29. [29]
    A. Pasquale and M. Sundari, Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type, Université de Grenoble. Annales de l’Institut Fourier 62 (2012), 859–886.Google Scholar
  30. [30]
    W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill Book Co., New York, 1987.Google Scholar
  31. [31]
    R. P. Sarkar and J. Sengupta, Beurling’s theorem and characterization of heat kernel for Riemannian symmetric spaces of noncompact type, Canadian Mathematical Bulletin 50 (2007), 291–312.Google Scholar
  32. [32]
    E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, Vol. 2, Princeton University Press, Princeton, NJ, 2003.Google Scholar
  33. [33]
    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, Vol. 32, Princeton University Press, Princeton, NJ, 1971.Google Scholar
  34. [34]
    A. Terras, Harmonic Analysis on Symmetric Spaces and Applications. II, Springer-Verlag, Berlin, 1988.Google Scholar
  35. [35]
    S. Thangavelu, An Introduction to the Uncertainty Principle, Progress in Mathematics, Vol. 217, Birkhäuser Boston, Inc., Boston, MA, 2004.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Stat-Math Unit, Indian Statistical InstituteKolkataIndia

Personalised recommendations