Surjectivity of convolution operators on harmonic $NA$ groups

TL;DR

This paper characterizes when convolution operators on harmonic $NA$ groups are surjective, linking this property to the slow decrease of the spherical Fourier transform, with applications to spherical averages.

Contribution

It establishes a precise criterion for the surjectivity of convolution operators on harmonic $NA$ groups based on Fourier transform properties.

Findings

Convolution operators are surjective iff their Fourier transform is slowly decreasing.

Spherical averages are shown to be surjective on smooth radial functions.

Provides a characterization connecting harmonic analysis and operator surjectivity.

Abstract

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