Distributional Extension and Invertibility of the $k$-Plane Transform and Its Dual

We investigate the distributional extension of the $k$-plane transform in $\mathbb{R}^d$ and of related operators. We parameterize the $k$-plane domain as the Cartesian product of the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^d$ with $\mathbb{R}^{d-k}$. This parameterization imposes an isotropy condition on the range of the $k$-plane transform which is analogous to the even condition on the range of the Radon transform. We use our distributional formalism to investigate the invertibility of the dual $k$-plane transform (the "backprojection" operator). We provide a systematic construction (via a completion process) to identify Banach spaces in which the backprojection operator is invertible and present some prototypical examples. These include the space of isotropic finite Radon measures and isotropic $L^p$-functions for $1 < p < \infty$. Finally, we apply our results to study a new form of regularization for inverse problems.