Range characterization of Radon transforms on complex projective spaces (Q1202856)

This paper deals with the range characterization of Radon transforms on complex projective spaces. The approach is based on the idea of F. John which yields a characterization by a single invariant differential operator on a Grassmann manifold. In fact, the range-characterizing operator can be represented as an ultra-hyperbolic differential operator like \(P=\partial^ 2/\partial \alpha_ 1\partial\beta_ 2 -\partial^ 2/\partial\alpha_ 2 \partial\beta_ 1)\) on a vector bundle, but the author here treats the one reduced to a single differential operator for the sake of simplicity. The main theorem of this paper is as follows: There exists a fourth order invariant differential operator \(P\) on \(M\) such that the range of the Radon transform \(R\) is characterized by \(P\), i.e. \(\text{Ker }P=\text{Im }R\).