Optimal bounds for the Hardy operator minus the Identity

We survey some recent works ([3,11,13]) concerning the study of the norm of the Hardy operator minus the Identity. In particular, optimal estimates are found for the cone of radially decreasing functions on the minimal Lorentz spaces Λ(X) (restricted type estimates), and a new Banach function space, closely related to Λ(X), is also naturally defined. We also mention a recent solution to a conjecture of Kruglyak and Setterqvist.