NOTE ON K-PLANE INTEGRAL TRANSFORMS

Let Π be a k-dimensional subspace of Rn, n ≥ 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π⊥. The k-plane transform of a measurable function f{hook} in the direction Π at the point x″ is defined by Lf{hook}(Π, x″) = ∝Πf{hook}(x′, x″) dx′. In this article certain a priori inequalities are established which show in particular that if f{hook} ε{lunate} Lp(Rn), 1 ≤ p $ ̌ n k, then f{hook} is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces Lp(Rn), p ≤ 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rn are also obtained. © 1979.