2024/07/14 by С. В. Асташкин, Astashkin, Sergey V. · 1 citation
Mathematics · #30H10 #30J05 #46A55 #46B22 #46E30 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2407.10178
openalex publication_date 2024/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that every measurable function f: [0,a]→ℂ 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 the absolute value of 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)).