Harmonic analysis associated with a discrete Laplacian

  1. Ciaurri, Ó. 2
  2. Alastair Gillespie, T. 3
  3. Roncal, L. 2
  4. Torrea, J.L. 1
  5. Varona, J.L. 2
  1. 1 Universidad Autónoma de Madrid

    Universidad Autónoma de Madrid

    Madrid, España

    GRID grid.5515.4

  2. 2 Universidad de La Rioja

    Universidad de La Rioja

    Logroño, España

    GRID grid.119021.a

  3. 3 University of Edinburgh

    University of Edinburgh

    Edimburgo, Reino Unido

    GRID grid.4305.2

Journal d'Analyse Mathematique

ISSN: 0021-7670

Year of publication: 2017

Volume: 132

Issue: 1

Pages: 109-131

Type: Article

Export: RIS
DOI: 10.1007/s11854-017-0015-6 SCOPUS: 2-s2.0-85021417820 WoS: 000404532200004 GOOGLE SCHOLAR


Cited by

  • Scopus Cited by: 26 (20-09-2021)

Journal Citation Reports

  • Year 2017
  • Journal Impact Factor: 0.592
  • Best Quartile: Q3
  • Area: MATHEMATICS Quartile: Q3 Rank in area: 206/310 (Ranking edition: SCIE)

SCImago Journal Rank

  • Year 2017
  • SJR Journal Impact: 1.262
  • Best Quartile: Q1
  • Area: Analysis Quartile: Q1 Rank in area: 28/148
  • Area: Mathematics (miscellaneous) Quartile: Q1 Rank in area: 54/439


  • Year 2017
  • CiteScore of the Journal : 1.6
  • Area: Mathematics (all) Percentile: 71
  • Area: Analysis Percentile: 60


It is well known that the fundamental solution of ut(n,t)=u(n+1,t)−2u(n,t)+u(n−1,t),n∈ℤ, with u(n, 0) = δnm for every fixed m ∈ Z is given by u(n, t) = e−2tIn−m(2t), where Ik(t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series Wtf(n) = Σm∈Ze−2tIn−m(2t)f(m). This formula allows us to analyze some operators associated with the discrete Laplacian using semigroup theory. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ℓp(Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. We also show that the Riesz transforms essentially coincide with the so-called discrete Hilbert transform defined by D. Hilbert at the beginning of the twentieth century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions. © 2017, Hebrew University Magnes Press.