Weak-Type Estimates for the Metaplectic Representation Restricted to the Shearing and Dilation Subgroup of \(SL(2,\mathbb {R})\)

  • Allesandra CauliEmail author
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We consider the subgroup G of \(SL(2,\mathbb {R})\) consisting of shearing and dilations, and we study the decay at infinity of the matrix coefficients of the metaplectic representation restricted to G. We prove weak-type estimates for such coefficients, which are uniform for functions in the modulation space \(M^1\). This work represents a continuation of a project aiming at studying weak-type and Strichartz estimates for unitary representations of non-compact Lie groups.



I am very indebted to Professors Fabio Nicola and Elena Cordero for discussions and remarks which improved this paper in an essential way.


  1. 1.
    A. Cauli, F. Nicola, A. Tabacco Strichartz estimates for the metaplectic representation, Rev. Mat. Iberoamer., to appear.Google Scholar
  2. 2.
    E. Cordero, F. De Mari, K. Nowak, A. Tabacco, Analytic features of reproducing groups for the metaplectic representation, J. Fourier Anal. Appl. 12, no. 2, 157–180, (2006).MathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Cordero, F. De Mari, K. Nowak, A. Tabacco, Reproducing groups for the metaplectic representation, Pseudo-differential operators and related topics, 227–244, Oper. Theory Adv. Appl., 164, Birkhäuser, Basel, (2006).Google Scholar
  4. 4.
    E. Cordero, F. De Mari, K. Nowak, A. Tabacco, Dimensional upper bounds for admissible subgroups for the metaplectic representation, Math. Nachr. 283, no. 7, 982–993 (2010).Google Scholar
  5. 5.
    E. Cordero, A. Tabacco, Triangular subgroups of Sp(d,R) and reproducing formulae, J. Funct. Anal. 264, no. 9, 2034–2058, (2013).MathSciNetCrossRefGoogle Scholar
  6. 6.
    E. Cordero, K. Gröchenig, F. Nicola, L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys. 55, no. 8, 081506 (2014).MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. Cordero, F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations 245, no. 7, 1945-1974, (2008).MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Cordero, F. Nicola, Strichartz estimates in Wiener amalgam space and applications to the Schrödinger equation, Math. Nachr. 281, no. 1, 25-41, (2008).MathSciNetCrossRefGoogle Scholar
  9. 9.
    E. Cordero, F. Nicola, Metaplectic representation on Wiener amalgam space and applications to the Schrödinger equation, J. Funct. Anal. 254, no.2, 506-534, (2008).MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples. Analyse harmonique sur les groupes de Lie (Nancy-Strasbourg, France, 1976–78), II, Lecture Notes in Math. 739, Springer, Berlin, 132–178, (1979).Google Scholar
  11. 11.
    M. Cowling, Herz’s “principe de majoration" and the Kunze-Stein phenomenon. Harmonic analysis and number theory (Montreal, PQ, 1996), 73–88, CMS Conf. Proc., 21, Amer. Math. Soc., (1997).Google Scholar
  12. 12.
    M. Cowling, U. Haagerup, R. Howe, Almost \(L^2\)matrix coefficient, J. Reine Angew. Math. 387, 97–110 (1988).Google Scholar
  13. 13.
    F. De Mari, K. Nowak Analysis of the affine transformations of the time-frequency plane, Bull. Austral. Math. Soc., Vol. 63, 195–218, (2001).MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. A. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Birkhäuser Basel, (2011).Google Scholar
  15. 15.
    M. A. de Gosson, The Wigner Transform, Advanced Textbooks in Mathematics, World Scientific, (2017).Google Scholar
  16. 16.
    K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Basel, (2001).Google Scholar
  17. 17.
    T. Tao, Nonlinear dispersive equations: local and global analysis, American Mathematical Society, (2006).Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly

Personalised recommendations