  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.
