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We prove that all convolution products of pairs of continuous orbitalmeasures in rank one, compact symmetric spaces are absolutely continuous anddetermine which convolution products are in $L^{2}$ (meaning, their densityfunction is in $L^{2})$. Characterizations of the pairs whose convolutionproduct is either absolutely continuous or in $L^2$ are given in terms of thedimensions of the corresponding double cosets. In particular, we prove that if$G/K$ is not $SU(2)/SO(2),$ then the convolution of any two regular orbitalmeasures is in $L^{2}$, while in $SU(2)/SO(2)$ there are no pairs of orbitalmeasures whose convolution product is in $L^{2}$.

Smoothness of convolution products of orbital measures on rank one compact symmetric spaces