Riesz Potentials and Orthogonal Radon Transforms on Affine Grassmannians

Abstract

Abstract We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j,k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j-dimensional affine planes in ℝ n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The main results include sharp existence conditions of 𝓡 j,k f on L p -functions, Fuglede type formulas connecting 𝓡 j,k with Radon-John k-plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡 j,k f under the assumption that f belongs to the range of the j-plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.