Surjectivity of Convolution Operators on Noncompact Symmetric Spaces

Let $\mu$ be a $K$-invariant compactly supported distribution on a noncompact Riemannian symmetric space $X=G/K$. If the spherical Fourier transform $\widetilde\mu(\lambda)$ is slowly decreasing, it is known that the right convolution operator $c_\mu\colon f\mapsto f*\mu$ maps $\mathcal E(X)$ onto $\mathcal E(X)$. In this paper, we prove the converse of this result. We also prove that $c_\mu$ has a fundamental solution if and only if $\widetilde\mu(\lambda)$ is slowly decreasing.