A Fractional Laplacian and Its Extension Problem

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