The smoothness of convolutions of zonal measures on compact symmetric spaces

We prove that for every irreducible, compact symmetric space, G c / K , of rank r , the convolution of any ( 2 r + 1 ) continuous, K -bi-invariant measures is absolutely continuous with respect to the Haar measure on G c . 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 G c is absolutely continuous with respect to Lebesgue measure. These results are nearly sharp.