Riesz transforms and fractional integration for orthogonal expansions on spheres, balls and simplices

This paper studies the Hardy–Littlewood–Sobolev (HLS) inequality and the Riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres, balls and simplexes with weights being invariant under a general finite reflection group on RdRd. The sharp index for the validity of the HLS inequality is determined and the LpLp-boundedness of the Riesz transforms is established. In particular, our results extend a classical inequality of Muckenhoupt and Stein on conjugate ultraspherical polynomial expansions. Our idea is based on a new decomposition of the Dunkl–Laplace–Beltrami operator on the sphere and some sharp asymptotic estimates of the weighted Christoffel functions.