A Fractional Laplacian and Its Extension Problem

In this paper, we establish four equivalent characterizations of the fractional Laplacian operator ${(-\Vert x\Vert {\Delta }_k)}^{\sigma }$ with $0<\sigma <1$, in some class of functions on ${ℝ}^d$. Here, ${\Delta }_k$ denotes the Dunkl differential-difference Laplacian and $k$ is a multiplicity function for the Dunkl operators. Starting from the natural Fourier characterization of ${(-\Vert x\Vert {\Delta }_k)}^{\sigma }$, we prove Bochner's and singular integral characterizations of ${(-\Vert x\Vert {\Delta }_k)}^{\sigma }$. A large part of the paper is dedicated to the fourth characterization where we obtain the fractional Laplacian ${(-\Vert x\Vert {\Delta }_k)}^{\sigma }$ as a Dirichlet-to-Neumann map via an extension problem to the upper half-plane. We also get a Poisson formula for the extension.