Radon transforms on affine Grassmannians

Abstract:

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