Radon transforms on affine Grassmannians

We develop an analytic approach to the Radon transform (f) over cap(zeta)=integral(tausubset ofzeta) f(tau), where f(tau) is a function on the affine Grassmann manifold G(n, k) of k-dimensional planes in R-n, and zeta is a k'-dimensional plane in the similar manifold G(n, k'), k'>k. For f is an element of L-p(G(n, k)), we prove that this transform is finite almost everywhere on G(n, k') if and only if 1less than or equal top < (n-k)/(k'-k), and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of Rn+1. It is proved that the dual Radon transform can be explicitly inverted for k+k'>= n-1, and interpreted as a direct, "quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if k+k'=n-1. The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.