Fulton B. Gonzalez

Radon Transforms on Grassmann Manifolds

If n = p + q + 1, the affine Grassmann manifolds G(p, n) of p-planes and G(q, n) of q-planes in Rn are dual homogeneous spaces of the Euclidean motion group M(n), with ξ ϵ G(p, n) and η ϵ G(q, n) being incident if and only if ξ meets η at a right angle. The resulting integral transform from functions on G(p, n) to functions on G(q, n) generalizes both the classical Radon transform and its dual. We show that this transform intertwines the actions of certain M(n)-invariant differential operators on G(p, n) and G(q, n), and we prove an inversion formula for this transform when n is odd, generalizing the Radon inversion formula, and obtaining in particular an inversion formula for the dual transform.

Invariant differential operators on Grassmann manifolds

For a Lie group G and a closed subgroup H c G let D( G/H) denote the algebra of differential operators D on the manifold G/H which are invariant under the action of G. In this note we determine this algebra in the case when G is the group M(n) of rigid motions of the Euclidean space R” and H is the subgroup of M(n) leaving a certain p-dimensional plane in R” invariant. Here 0 6 p 6 II - 1 and is the space

Support theorems for Radon transforms on higher rank symmetric spaces

We prove support theorems for Radon transforms with real-analytic measures on horocycles in higher rank symmetric spaces. The microlocal analysis is more difficult than for rank one, but we prove a generalization of Helgasons support theorem and a theorem that is new even in the classical case.

Dual Radon transforms on affine Grassmann manifolds

Fix 0 ≤ p 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.