Range characterization of Radon transforms on \(S^ n\) and \(P^ n\mathbb{R}\) (Q1321784)

Let be \(R = R_ \ell\) the Radon transform on the \(n\)-dimensional sphere \(S^ n\), \((1 \leq \ell \leq n - 2\)), \(f\) an integrating function on \(S^ n\) and \(\xi \subset S^ n\) an oriented \(\ell\)-dimensional totally geodesic sphere. One defines \(R\) as follows \[ Rf(\xi) = {1\over \text{Vol}(S^ \ell)} \int_{x \in \xi} f(x) dv_ \xi(x), \] where \(dv_ \xi(x)\) is the canonical measure on \(\xi\). In this paper, the author proves the following theorem: There exists a fourth order invariant differential operator \(P\) on \(\widetilde{Gr}_{\ell + 1,n + 1}\) such that the range \(\text{Im }R\) of \(R\) is identical with the kernel \(\text{Ker }P\).