Dual Radon transforms on affine Grassmann manifolds

Abstract

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