Surjectivity of Convolution Operators on Noncompact Symmetric Spaces

Abstract

Let $\mu$ be a $K$-invariant compactly supported distribution on a noncompact Riemannian symmetric space $X=G/K$. If the spherical Fourier transform $\tilde{\mu }(\lambda )$ is slowly decreasing, it is known that the right convolution operator $c_{\mu }:f\mapsto f\ast \mu$ 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 $\tilde{\mu }(\lambda )$ is slowly decreasing.