Radon Transforms on Affine Grassmannians

We develop an analytic approach to the Radon transform f̂(ζ) = ∫ τ⊂ζ f(τ), where f(τ) is a function on the affine Grassmann manifold G(n, k) of k-dimensional planes in Rn, and ζ is a k′-dimensional plane in the similar manifold G(n, k′), k′ > k. For f ∈ Lp(G(n, k)), we prove that this transform is finite almost everywhere on G(n, k′) if and only if 1 ≤ p < (n− k)/(k′− k), and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of Rn+1. It is proved that the dual Radon transform can be explicitly inverted for k + k′ ≥ n − 1, and interpreted as a direct, “quasi-orthogonal” Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if k + k′ = n−1. The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.