A Fractional Laplacian and Its Extension Problem

ABSTRACT In this paper, we establish four equivalent characterizations of the fractional Laplacian operator with , in some class of functions on . Here, denotes the Dunkl differential‐difference Laplacian and is a multiplicity function for the Dunkl operators. Starting from the natural Fourier characterization of , we prove Bochner's and singular integral characterizations of . A large part of the paper is dedicated to the fourth characterization where we obtain the fractional Laplacian as a Dirichlet‐to‐Neumann map via an extension problem to the upper half‐plane. We also get a Poisson formula for the extension.