Title: Existence and non-existence of extremizers for certain k-plane transform inequalities
Abstract: The k-plane transform $R$ satisfies the inequality $|Rf|_{L^q} \leq C |f|_{L^p}$ for certain exponents $p,q$. We review how the presence of a hidden conformal symmetry (found originally by Drury then rediscovered by Christ) has led to complete results on existence, non-existence and uniqueness of extremizers in the endpoint case. If time permits we will discuss the issue appearing when considering the non-endpoint cases. This talk is based on work by Christ, Drouot, Drury and Flock.