Fractional Laplacian on the torus

Abstract

We study the fractional Laplacian [Formula: see text] on the [Formula: see text]-dimensional torus [Formula: see text], [Formula: see text]. First, we present a general extension problem that describes any fractional power [Formula: see text], [Formula: see text], where [Formula: see text] is a general nonnegative self-adjoint operator defined in an [Formula: see text]-space. This generalizes to all [Formula: see text] 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 [Formula: see text], when [Formula: see text]. 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.