The C∞-isomorphism property for a class of singularly-weighted X-ray transforms

Abstract

We study various self-adjoint realizations of the X-ray transform on the Euclidean disk D, obtained by considering specific singularly weighted L2 topologies. We first recover the well-known Singular Value Decompositions in terms of disk orthogonal polynomials, then prove that each such realization is an isomorphism of C?(D). As corollaries: we give some range characterizations; we explain how for such choices, these normal operators are in the functional calculus of two distinguished differential operators; we show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces.