Range theorems and inversion formulas for Radon transforms on Grassmann manifolds (Q1385026)

Let \(G_p =\text{Gr}(p,n;{\mathbb F})\) be the Grassmann manifold of all \(p\)-dimensional subspaces in \({\mathbb F} ^n\), for \(\mathbb F\) the real or complex number field. Let \(r(p) = \text{min}\{p,n-p\}\) be its rank. The Radon transform \(R_p ^q \colon C^\infty (G_q) \to C^\infty (G_p)\) is known to be injective when \(r(q)\leq r(p)\). In the paper under review, the author characterizes the range of \(R _p ^q\) as the kernel of an invariant differential operator. Also, explicit inversion formulas for the Radon transform are given. The results are stated without proof.