We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star \delta_{e^Y}^\natural$ of two orbital measures on the symmetric space ${\bf SO}0(p,q)/{\bf SO}(p)\times{\bf SO}(q)$, $q>p$. We prove sharp conditions on $X$, $Y\in\a$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for $\SO_0(p,q)/\SO(p)\times\SO(q)$ will also serve for the spaces ${\bf SU}(p,q)/{\bf S}({\bf U}(p)\times{\bf U}(q))$ and ${\bf Sp}(p,q)/{\bf Sp}(p)\times{\bf Sp}(q)$, $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $\delta{e^X}^\natural$.
