Dual radon transforms on affine grassmann manifolds

Fix 0 ≤ p < q ≤ n - 1, and let G(p, n) and G(q, n) denote the affine Grassmann manifolds of p- and q-planes in ℝ ⁿ. We investigate the Radon transform R ^(q,p) : C ^∞(G(q, n)) → C ^∞(G(p, n)) associated with the inclusion incidence relation. For the generic case dim G(q, n) < dim G(p, n) and p + q > n, we will show that the range of this transform is given by smooth functions on G(p, n) annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case p + q = n.