Fractional Laplacian on the torus

  1. Roncal, L. 1
  2. Stinga, P.R. 2
  1. 1 Universidad de La Rioja

    Universidad de La Rioja

    Logroño, España

    GRID grid.119021.a

  2. 2 University of Texas at Austin

    University of Texas at Austin

    Austin, Estados Unidos

    GRID grid.89336.37

Communications in Contemporary Mathematics

ISSN: 0219-1997

Year of publication: 2016

Volume: 18

Issue: 3

Type: Article

Export: RIS
DOI: 10.1142/S0219199715500339 SCOPUS: 2-s2.0-84961696296 WoS: 000373281200001 GOOGLE SCHOLAR


Cited by

  • Scopus Cited by: 36 (12-06-2021)

Journal Citation Reports

  • Year 2016
  • Journal Impact Factor: 1.191
  • Best Quartile: Q1
  • Area: MATHEMATICS Quartile: Q1 Rank in area: 42/311 (Ranking edition: SCIE)
  • Area: MATHEMATICS, APPLIED Quartile: Q2 Rank in area: 86/255 (Ranking edition: SCIE)

SCImago Journal Rank

  • Year 2016
  • SJR Journal Impact: 1.403
  • Best Quartile: Q1
  • Area: Applied Mathematics Quartile: Q1 Rank in area: 73/556
  • Area: Mathematics (miscellaneous) Quartile: Q1 Rank in area: 50/427


  • Year 2016
  • CiteScore of the Journal : 1.8
  • Area: Mathematics (all) Percentile: 79
  • Area: Applied Mathematics Percentile: 51


We study the fractional Laplacian (-δ)σ/2 on the n-dimensional torus n, n ≥ 1. First, we present a general extension problem that describes any fractional power Lγ, γ > 0, where L is a general nonnegative self-adjoint operator defined in an L2-space. This generalizes to all γ > 0 and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular, it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack's inequalities for (-δ)σ/2, when 0 < σ < 2. We deduce regularity estimates on Hölder, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach. © 2016 World Scientific Publishing Company.