Convolution of orbital measures in symmetric spaces

Let G/K be a noncompact symmetric space, G_c/K its compact dual, g = t ⊕ p the Cartan decomposition of the Lie algebra g of G, a a maximal abelian subspace of a, H be an element of a, a=exp (H) , and a_c =exp (iH). In this paper, we prove that if for some positive integer r, v _ac^r is absolutely continuous with respect to the Haar measure on G_c, then v^r_a is absolutely continuous with respect to the left Haar measure on G, where a_c (respectively a) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare ['Singular dichotomy for orbital measures on complex groups', Boll.Unione Mat.Ital.(9)III(2010), 409-419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.