Continuity properties of k-plane integrals and Besicovitch sets

Abstract

Let Π be a k-dimensional subspace of Rn(n ≥ 2) and let Π┴ denote its orthogonal complement. If x ∈ Rn we shall write x = x0 + x′ with x0 ∈ Πand x′ ∈ Π ┴. If f(x) is a real measurable function on Rn, the k-plane integral F(Π,x′ )is defined as the integral of f over the affine subspace Π + x′ with respect to k-dimensional Lebesgue measure (assuming that the integral exists). If k = 1 we get the x-ray transform that arises in the problem of radiographic reconstruction, and if k = n − 1, the k-plane integral is the usual projection or Radon transform. The paper by Smith, Solmon and Wagner (4) contains a survey of results on k-plane integrals. Here we shall be interested in the behaviour of the F (Π, x′ ) regarded as a function of x′ for fixed Π for various classes of function f. We shall obtain some surprisingly strong results on the continuity and differentiability of F(Π,x′) with respect to x′ for almost all Π (in the sense of the appropriate Haar measure). As will be seen the dimensions n and k have a crucial effect on what may be said, and most of our results will be confined to the cases where k > ½n.