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 $\mathbb{R}^d$.