Surjectivity of Convolution Operators on Noncompact Symmetric Spaces

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

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