Fractional Laplacian on the torus

2012/09/27 by L. Roncal, Roncal, L., P. R. Stinga +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.CA #math.FA

paper · pdf · doi:10.48550/arxiv.1209.6104

18 pages, 2 figures. To appear in Communications in Contemporary Mathematics

arxiv created 2015/01/28 · arxiv updated 2015/01/29

Abstract

We study the fractional Laplacian (-Δ)^σ/2 on the n-dimensional torus \mathbbTⁿ, n≥1. First, we present a general extension problem that describes any fractional power L^γ, γ>0, where L is a general nonnegative selfadjoint 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.

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