Littlewood-Paley Theory For Orthogonal Expansions Associated With Root Systems

We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley $g$-functions in the spirit of Stein’s work for compact Lie groups and prove their $L^p$-boundedness for $1<p<\mathrm{\infty }$. As an application, we define associated Riesz transforms and Riesz potentials and prove their $L^p$-continuity for $1<p<\mathrm{\infty }$. Passing to the average with respect to the action of the associated reflection group, our framework in particular covers the Littlewood-Paley-Stein theory for Jacobi polynomial expansions and corresponding direct product settings as special cases.