  In this paper we study mixed norm boundedness for fractional integrals
related to Laplace--Beltrami operators on compact Riemannian symmetric spaces
of rank one. The key point is the analysis of weighted inequalities for
fractional integral operators associated to trigonometric Jacobi polynomials
expansions. In particular, we find a novel sharp estimate for the Jacobi
fractional integral kernel with explicit dependence on the type parameters.
