Surjectivity of convolution operators on harmonic $NA$ groups

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