Surjectivity of Mean Value Operators on Noncompact Symmetric Spaces

Let \(X=G/K\) 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.