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We investigate the distributional extension of the $k$-plane transform in$\mathbb{R}^d$ and of related operators. We parameterize the $k$-plane domainas the Cartesian product of the Stiefel manifold of orthonormal $k$-frames in$\mathbb{R}^d$ with $\mathbb{R}^{d-k}$. This parameterization imposes anisotropy condition on the range of the $k$-plane transform which is analogousto the even condition on the range of the Radon transform. We use ourdistributional formalism to investigate the invertibility of the dual $k$-planetransform (the "backprojection" operator). We provide a systematic construction(via a completion process) to identify Banach spaces in which thebackprojection 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 anew form of regularization for inverse problems.

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