Moment Conditions and Support Theorems for Radon Transforms on Affine Grassmann Manifolds

Let $G(p,n)$ and $G(q,n)$ be the affine Grassmann manifolds of $p$- and $q$- planes in ${\mathbb R}^n$, respectively, and let $\mathcal{R}^{(p,q)}$ be the Radon transform from smooth functions on $G(p,n)$ to smooth functions on $G(q,n)$ arising from the inclusion incidence relation. When $p<q$ and $\dim G(p,n) = \dim G(p,n)$, we present a range characterization theorem for $\mathcal{R}^{(p,q)}$ via moment conditions. We then use this range result to prove a support theorem for $\mathcal{R}^{(p,q)}$. This complements a previous range characterization theorem for $\mathcal{R}^{(p,q)}$ via differential equations when $\dim G(p,n) < \dim G(p,n)$. We also present a support theorem in this latter case.