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-nu-smoothness of the Radon Nikodym derivative f(a1,...,ap) = d (mu(a1) *... * mu(ap)) / d mu(G) of the convolution mu(a1) *... * mu(ap) of some orbital measures mu(ai), with respect to the Haar measure mu(G) of G. This generalizes some of the main results in 1121, in the case of compact rank one symmetric spaces, where the absolute continuity of the measure it mu(a1) *... * mu(ap) with respect to d mu(G) was considered. Our main result generalizes also the main results in [1] and [7], where the L-2-regularity was considered. As a consequence of our main result, we give sufficient conditions for f(a1,...,ap) to be in L-q (G, d mu(G)) for all q >= 1 and for the Fourier series of f(a1,...,ap) to converge absolutely and uniformly to ff(a1,...,ap)(. )