Transference of fractional Laplacian regularity

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

    Universidad de La Rioja

    Logroño, España

    GRID grid.119021.a

  2. 2 University of Texas at Austin
    info

    University of Texas at Austin

    Austin, Estados Unidos

    GRID grid.89336.37

Book:
Special Functions, Partial Differential Equations, and Harmonic Analysis

ISBN: 9783319105444

Year of publication: 2014

Volume: 108

Pages: 203-212

Type: Book chapter

Export: RIS
DOI: 10.1007/978-3-319-10545-1_14 SCOPUS: 2-s2.0-84969228139 arXiv: 1406.6807 GOOGLE SCHOLAR lock_openOpen access editor

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Abstract

In this note we show how to obtain regularity estimates for the fractional Laplacian on the multidimensional torus Tn from the fractional Laplacian on ℝn. Though at first glance this may seem quite natural, it must be carefully precised. A reason for that is the simple fact that L2functions on the torus cannot be identified with L2functions on ℝn. The transference is achieved through a formula that holds in the distributional sense. Such an identity allows us to transfer Harnack inequalities, to relate the extension problems, and to obtain pointwise formulas and Hölder regularity estimates. © Springer International Publishing Switzerland 2014.