Radon transforms on affine Grassmannians

Abstract

We develop an analytic approach to the Radon transform f ^ ( ζ ) = ∫ τ ⊂ ζ f ( τ ) \hat f (\zeta )=\int _{\tau \subset \zeta } f (\tau ) , where f ( τ ) f(\tau ) is a function on the affine Grassmann manifold G ( n , k ) G(n,k) of k k -dimensional planes in R n \mathbb {R}^n , and ζ \zeta is a k ′ k’ -dimensional plane in the similar manifold G ( n , k ′ ) , k ′ > k G(n,k’), \; k’>k . For f ∈ L p ( G ( n , k ) ) f \in L^p (G(n,k)) , we prove that this transform is finite almost everywhere on G ( n , k ′ ) G(n,k’) if and only if 1 ≤ p > ( n − k ) / ( k ′ − k ) \; 1 \le p > (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 R n + 1 \mathbb {R}^{n+1} . It is proved that the dual Radon transform can be explicitly inverted for k + k ′ ≥ n − 1 k+k’ \ge 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 k+k’ = n-1 . The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.