Existence and non-existence of extremizers for certain $k$-plane transform inequalities

We provide sharp forms of $k$-plane transform inequalities on the $d$-dimensional sphere ${\mathbb{S}}^d$ and the $d$-dimensional hyperbolic space ${\mathbb{H}}^d$. In particular, we prove that extremizers do not exist for ${\mathbb{H}}^d$. This work is a natural extension of some results for the $k$-plane transform on ${\mathbb{R}}^d$.