On the kernel of the product formula on symmetric spaces

In this article, we study the regularity properties of the density of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We also give a geometric construction of its support S = a(e^X K e^Y) where e^a(g) is the abelian part in the Cartan decomposition of g. Using spherical Fourier theory, an expression for the kernel is given under certain conditions. Our approach also leads to some interesting conclusions for the kernel of the Abel transform.