On the set of extreme points of the unit ball of a Hardy-Lorentz space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917726354669568 |
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| author | Astashkin, Sergey V. |
| author_facet | Astashkin, Sergey V. |
| contents | 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 $Λ(φ)$ 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 $H^1$ showing that $H^1$ 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 $f\in H(Λ(φ))$, $\|f\|_{H(Λ(φ))}=1$, such that the absolute value of its nontangential limit ${f}(e^{it})$ is a constant on some set of positive measure of $[0,2π]$, is an extreme point of the unit ball of $H(Λ(φ))$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_10178 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the set of extreme points of the unit ball of a Hardy-Lorentz space Astashkin, Sergey V. Functional Analysis 46E30, 30H10, 30J05, 46A55, 46B22 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 $Λ(φ)$ 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 $H^1$ showing that $H^1$ 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 $f\in H(Λ(φ))$, $\|f\|_{H(Λ(φ))}=1$, such that the absolute value of its nontangential limit ${f}(e^{it})$ is a constant on some set of positive measure of $[0,2π]$, is an extreme point of the unit ball of $H(Λ(φ))$. |
| title | On the set of extreme points of the unit ball of a Hardy-Lorentz space |
| topic | Functional Analysis 46E30, 30H10, 30J05, 46A55, 46B22 |
| url | https://arxiv.org/abs/2407.10178 |