THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

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