Fractional Laplacian on the torus

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

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 University of Texas at Austin
    info

    University of Texas at Austin

    Austin, Estados Unidos

    ROR https://ror.org/00hj54h04

Journal:
Communications in Contemporary Mathematics

ISSN: 0219-1997

Year of publication: 2016

Volume: 18

Issue: 3

Type: Article

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

More publications in: Communications in Contemporary Mathematics

Abstract

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.