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

Abstract

We prove that every measurable function $f:\,[0,a]\to\mathbb{C}$ 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 not linear, strictly increasing, concave, continuous function on $[0,a]$ with $\varphi(0)=0$. 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 a unique Hardy-Lorentz space $H(\Lambda(\varphi))$, for which every extreme point of the unit ball is a normed outer function. Moreover, assuming that $\varphi$ is strictly increasing and strictly concave, we prove that every function $f\in H(\Lambda(\varphi))$, $\|f\|_{H(\Lambda(\varphi))}=1$, such that the absolute value of its nontangential limit ${f}(e^{it})$ is a constant on some set of positive measure of $[0,2\pi]$, is an extreme point of the unit ball of $H(\Lambda(\varphi))$.