Dual Radon transforms on affine Grassmann manifolds

Fix $0 \leq p < q \leq n-1$, and let $G(p,n)$ and $G(q,n)$denote the affine Grassmann manifolds of $p$- and $q$-planes in $\mathbb{R} ^n$. We investigate the Radon transform $\mathcal{R}^{(q,p)} : C^{\infty} (G(q,n)) \to C^{\infty} (G(p,n))$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 $p+q =n$.