We prove that every measurable function \(f:\,[0,a]\rightarrow \mathbb {C}\) such that \(|f|=1\) a.e. on [0, a] is an extreme point of the unit ball of the Lorentz space \(\Lambda (\varphi )\) on [0, a] whenever \(\varphi \) is a nonlinear, strictly increasing, concave, continuous function on [0, a] with \(\varphi (0)=0\) and \(\varphi (a)=1\) . As a consequence, we complement the classical de Leeuw–Rudin theorem on a description of extreme points of the unit ball of \(H^1\) showing that \(H^1\) is the unique Hardy–Lorentz space \(H(\Lambda (\varphi ))\) for which every extreme point of the unit ball is an outer function of norm 1. Moreover, assuming that \(\varphi \) is strictly increasing and strictly concave, we prove that every function \(f\in H(\Lambda (\varphi ))\) , \(\Vert f\Vert _{H(\Lambda (\varphi ))}=1\) , such that the nontangential limit \({f}(e^{it})\) has constant modulus on some set of positive measure of \([0,2\pi ]\) , is an extreme point of the unit ball of \(H(\Lambda (\varphi ))\) .
