Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

令 $G/K$ 为不可约对称空间，其中G是非紧连通李群，K是紧连通子群。我们使用球函数的衰减特性来证明任何 $r=r(G/K)$ 连续轨道测量的卷积乘积在 $L^{2}(G)$ 中具有密度函数，因此是绝对连续的相对于 Haar 度量的度量。数字r大约是 $G/K$ 的等级。对于轨道测量的特殊情况， $\nu _{a_{i}}$ 支持双陪集 $Ka_{i}K$ ，其中 $a_{i}$ 属于稠密的规则元素集，我们证明了 $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ 除了对称空间当需要三个轨道测量的卷积时，Cartan 类 $AI$ （即使 $\nu _{a_{1}}\ast \nu _{a_{2}}$ 是绝对连续的）。