The object of this study is an integral operator \mathcal{S} which averages functions in the Euclidean upper half-space \mathbb{R}_{+}^{n} over the half-spheres centered on the topological boundary \partial \mathbb{R}_{+}^{n}. By generalizing Norton's approach to the inversion of arc means in the upper half-plane, we intertwine \mathcal{S} with a convolution operator... Show more