The smoothness of convolutions of zonal measures on compact symmetric spaces

We prove that for every irreducible, compact symmetric space, Gc/K, of rank r, the convolution of any (2r+1) continuous, K-bi-invariant measures is absolutely continuous with respect to the Haar measure on Gc. We also prove that the convolution of (r+1) continuous, K-invariant measures on the −1 eigenspace in the Cartan decomposition of the Lie algebra of Gc is absolutely continuous with respect to Lebesgue measure. These results are nearly sharp.