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
    info

    Universidad Autónoma de Madrid

    Madrid, España

    ROR https://ror.org/01cby8j38

  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  3. 3 University of Edinburgh
    info

    University of Edinburgh

    Edimburgo, Reino Unido

    ROR https://ror.org/01nrxwf90

Revista:
Journal d'Analyse Mathematique

ISSN: 0021-7670

Año de publicación: 2017

Volumen: 132

Número: 1

Páginas: 109-131

Tipo: Artículo

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DOI: 10.1007/S11854-017-0015-6 SCOPUS: 2-s2.0-85021417820 WoS: WOS:000404532200004 GOOGLE SCHOLAR

Otras publicaciones en: Journal d'Analyse Mathematique

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Resumen

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.