The smoothness of convolutions of zonal measures on compact symmetric spaces (Q1947328)

Let \(G_c/K\) be an irreducible Riemannian space of compact type. Denote by \(r\) the rank of \(G_c/K\). In the paper under review, the authors prove that the convolution of any \(2r+1\) continuous and \(K\)-bi-invariant measures is absolutely continuous with respect to the Haar measure on \(G_c\). In addition, they 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 the Lebesgue measure. The authors show the exactness of these results and note that the ideas were inspired by a paper by A. Wright, 2011.