Fractional Laplacian on the torus

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.