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 $\Lambda (\phi )$ on $[0,a]$ whenever $\phi$ is a not linear, strictly increasing, concave, continuous function on $[0,a]$ with $\phi (0)=0$. 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 $H(\Lambda (\phi ))$, for which every extreme point of the unit ball is a normed outer function. Moreover, assuming that $\phi$ is strictly increasing and strictly concave, we prove that every function $f\in H(\Lambda (\phi ))$, $\Vert f{\Vert }_{H(\Lambda (\phi ))}=1$, 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 $H(\Lambda (\phi ))$.

submitted