On the set of extreme points of the unit ball of a Hardy-Lorentz space

We prove that every measurable function f:\,[0,a] such that $|f|=1$ a.e. on $[0,a]$ is an extreme point of the unit ball of the Lorentz space Λ(φ) on $[0,a]$ whenever φ is a not linear, strictly increasing, concave, continuous function on $[0,a]$ with φ(0)=0. As a consequence, we complement the classical de Leeuw-Rudin theorem on a description of extreme points of the unit ball of H¹ showing that H¹ is a unique Hardy-Lorentz space H(Λ(φ)), for which every extreme point of the unit ball is a normed outer function. Moreover, assuming that φ is strictly increasing and strictly concave, we prove that every function f∈ H(Λ(φ)), \|f\|H(Λ(φ))=1, such that its nontangential limit f(eⁱᵗ) is a constant on some set of positive measure of [0,2π], is an extreme point of the unit ball of H(Λ(φ)).