A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)

  • 1 Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India
Tapendu RanaORCID iD: https://orcid.org/0000-0002-0406-1869

Abstract

In this paper, we prove a genuine analogue of the Wiener Tauberian theorem for Lp,1(G) (1p<2), with G=SL(2,).

  • [1]

    R. A. Askey, T. H. Koornwinder and W. Schempp, Special Functions: Group Theoretical Aspects and Application, Springer, Cham, 1984.

  • [2]

    W. H. Barker, L p {L^{p}} harmonic analysis on SL ( 2 , ) {\mathrm{SL}(2,\mathbb{R})}, Mem. Amer. Math. Soc. 393 (1998), 1–110.

  • [3]

    Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit, Wiener’s Tauberian theorem for spherical functions on the automorphism group of the unit disk, Ark. Mat. 34 (1996), no. 2, 199–224.

    • Crossref
    • Export Citation
  • [4]

    Y. Benyamini and Y. Weit, Harmonic analysis of spherical functions on SU(1, 1), Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 671–694.

    • Crossref
    • Export Citation
  • [5]

    R. Camporesi, The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces, J. Lie Theory 7 (1997), 29–60.

  • [6]

    T. Carleman, L ’integrale de Fourier et questions qui s’y rattachent, Almqvist & Wiksells, Uppsala, 1944.

  • [7]

    M. Cowling, The Kunze–Stein phenomenon, Ann. of Math. (2) 107 (1978), no. 2, 209–234.

    • Crossref
    • Export Citation
  • [8]

    G. van Dijk, Introduction to Harmonic Analysis and Generalized Gelfand Pairs, De Gruyter Stud. Math. 36, Walter de Gruyter, Berlin, 2009.

  • [9]

    Y. Domar, On the analytic transform of bounded linear functionals on certain Banach algebras, Studia Math. 53 (1975), no. 3, 203–224.

    • Crossref
    • Export Citation
  • [10]

    A. Dahlner, A Wiener Tauberian theorem for weighted convolution algebras of zonal functions on the automorphism group of the unit disc, Bergman Spaces and Related Topics in Complex Analysis, Contemp. Math. 404, American Mathematical Society, Providence (2006), 67–102.

  • [11]

    L. Ehrenpreis and F. I. Mautner, Some properties of the Fourier transform on semisimple Lie groups. I, Ann. of Math. (2) 61 (1955), 406–439.

    • Crossref
    • Export Citation
  • [12]

    L. Ehrenpreis and F. I. Mautner, Some properties of the Fourier transform on semisimple Lie groups. III, Trans. Amer. Math. Soc. 90 (1959), 431–484.

  • [13]

    A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill, New York, 1953.

  • [14]

    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Elsevier, Amsterdam, 2007.

  • [15]

    L. Grafakos, Classical Fourier Analysis, 2nd ed., Springer, New York, 2008.

  • [16]

    S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Progr. Math. 13, Birkhäuser, Boston, 1981.

  • [17]

    N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Englewood Cliffs, 1965.

  • [18]

    E. K. Narayanan, Wiener Tauberian theorems for L 1 ( K G / K ) {L^{1}(K\setminus G/K)}, Pacific J. Math. 241 (2009), no. 1, 117–126.

    • Crossref
    • Export Citation
  • [19]

    E. K. Narayanan and A. Sitaram, Analogues of the Wiener Tauberian and Schwartz theorems for radial functions on symmetric spaces, Pacific J. Math. 249 (2011), no. 1, 199–210.

    • Crossref
    • Export Citation
  • [20]

    S. Pusti, S. K. Ray and R. P. Sarkar, Wiener–Tauberian type theorems for radial sections of homogeneous vector bundles on certain rank one Riemannian symmetric spaces of noncompact type, Math. Z. 269 (2011), no. 1–2, 555–586.

    • Crossref
    • Export Citation
  • [21]

    S. Pusti and A. Samanta, Wiener Tauberian theorem for rank one semisimple Lie groups and for hypergeometric transforms, Math. Nachr. 290 (2017), no. 13, 2024–2051.

    • Crossref
    • Export Citation
  • [22]

    R. P. Sarkar, Wiener Tauberian theorems for SL ( 2 , ) {\mathrm{SL}(2,\mathbb{R})}, Pacific J. Math. 177 (1997), no. 2, 291–304.

    • Crossref
    • Export Citation
  • [23]

    R. P. Sarkar, Wiener Tauberian theorem for rank one symmetric spaces, Pacific J. Math. 186 (1998), no. 2, 349–358.

    • Crossref
    • Export Citation
  • [24]

    A. Sitaram, On analogue of the Wiener Tauberian theorem for symmetric spaces of the noncompact type, Pacific J. Math. 133 (1988), 197–208.

    • Crossref
    • Export Citation
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