Fractional Laplacian on the torus
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1
Universidad de La Rioja
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2
University of Texas at Austin
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ISSN: 0219-1997
Année de publication: 2016
Volumen: 18
Número: 3
Type: Article
beta Ver similares en nube de resultadosD'autres publications dans: Communications in Contemporary Mathematics
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Résumé
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.