Relatively closed operators associated with a pair of Grassmannians.
Given an incidence relation \(A \subset X \times Y\), it is said that a linear differential operator \(\kappa\) acting on functions on \(Y\) and taking values in differential forms on \(Y\) is relatively closed if it satisfies the relation \(d \kappa(Jf) |_{Y^x} =0\). Here \(f\) is an arbitrary function on \(Y\), \(Jf\) is the integral of \(f\) over \(X^y = (X,y) \cap A\), and \(Y^x = (x,Y) \cap A\). Such operators naturally arise in integral geometry defining local inversion formulas. In the present paper the author considers the case when \(X\) and \(Y\) are Grassmannians and \(A\) is the flag manifold, constructs families of such operators and describes all local inversion formulas for these transformations.
scientific article; zbMATH DE number 1928148