Smoothness of the Radon-Nikodym derivative of a convolution of orbital measures on compact symmetric spaces of rank one

Let G/K be a compact symmetric space of rank one. The aim of this paper is to give sufficient conditions for the C_v -smoothness of the Radon Nikodym derivative f_a1,...,ap = d (μ_a1 * ... * μ_ap of the convolution μ_a1 *...*μ_ap of some orbital measures μ_ai, with respect to the Haar measure μG of G. This generalizes some of the main results in [12], in the case of compact rank one symmetric spaces, where the absolute continuity of the measure μ_a1 * ... * μ_ap with respect to dμG was considered. Our main result generalizes also the main results in [1] and [7], where the L²-regularity was considered. As a consequence of our main result, we give sufficient conditions for f_a1,...,ap to be in Lq (G, dμG) for all q ge; 1 and for the Fourier series of f_a1,...,ap to converge absolutely and uniformly to f_a1,...,ap.