Fractional Integration and Fractional Differentiation for d-dimensional Jacobi Expansions

Abstract

In this paper we consider an alternative orthogonal decomposition of the space $L^2$ associated to the $d$-dimensional Jacobi measure and obtain an analogous result to P.A. Meyer's Multipliers Theorem for d-dimensional Jacobi expansions. Then we define and study the Fractional Integral, the Fractional Derivative and the Bessel potentials induced by the Jacobi operator. We also obtain a characterization of the potential spaces and a version of Calderon's reproduction formula for the d-dimensional Jacobi measure.