Surjectivity of Convolution Operators on Harmonic NA Groups

Let \(\mu \) $\mu$ be a radial compactly supported distribution on a harmonic NA group. We prove that the right convolution operator \(c_{\mu }:f \mapsto f* \mu \) $c_{\mu }:f\mapsto f\ast \mu$ maps the space of smooth \(\mathfrak {v}\) $\mathfrak{v}$ -radial functions onto itself if and only if the spherical Fourier transform \(\widetilde{\mu }(\lambda )\) $\tilde{\mu }(\lambda )$ , \(\lambda \in \mathbb {C}\) $\lambda \in \mathbb{C}$ , is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth \(\mathfrak {v}\) $\mathfrak{v}$ -radial functions.