Surjectivity of Mean Value Operators on Noncompact Symmetric Spaces

Let $X=G\mathrm{/}K$ be a symmetric space of the non-compact type. We prove that the mean value operator over translated $K$-orbits of a fixed point is surjective on the space of smooth functions on $X$ if $X$ is either complex or of rank one. For higher rank spaces it is shown that the same statement is true for points in an appropriate Weyl subchamber.