k-Plane integral transforms

Let II be a k-dimensional subspace of R/sup n/, n greater than or equal to 2, and write x = (x',x'') with x' in II and x'' in the orthogonal complement II/sup perpendicular to/. The k-plane transform of a measurable function f in the direction II at the point x'' is defined by Lf(II,X'' = f/sub II/f(x',x'') dx'. Certain a priori inequalities are established that show in particular that, if f is an element of L/sup p/(R/sup n/), 1 less than or equal to p < n/k, then f is integrable over almost every translate of almost every k space. Mapping properties of the k-plane transform between the spaces L/sup p/(R/sup n/), p less than or equal to 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in R/sup n/ are also obtained