A Littlewood-Type Theorem for Random Bergman Functions
A Littlewood-Type Theorem for Random Bergman Functions
复制标题
随机Bergman函数的Littlewood型定理
DOI：
10.1093/imrn/rnab018
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发表时间：
2021-04
影响因子：
1
通讯作者：
Chao Liu
中科院分区：
文献类型：
--
作者：
Guozheng Cheng;Xiang Fang;Chao Liu
Abstract. Let $f(z)=sum _{n=0}^{infty }a_n z^n$ be a formal power series with complex coefficients. Let $({mathcal{R}} f)(z)= sum _{n=0}^{infty }pm a_n z^n$ be the randomization of $f$ by choosing independently a random sign for each coefficient. Let $H^p({mathbb{D}})$ and $L^p_a({mathbb{D}})$ $(p0)$ denote the Hardy space and the Bergman space, respectively, over the unit disk in the complex plane. In 1930, Littlewood proved that if $f in H^2({mathbb{D}})$, then ${mathcal{R}} f in H^p({mathbb{D}})$ for any $p in (0, infty )$ almost surely, and if $f notin H^2({mathbb{D}})$, then ${mathcal{R}} f notin H^p({mathbb{D}})$ for any $p in (0, infty )$ almost surely. In this paper, we obtain a characterization of the pairs $(p, q) in (0, infty )^2$ such that ${mathcal{R}} f$ is almost surely in $L^q_a({mathbb{D}})$ whenever $f in L^p_a({mathbb{D}})$, including counterexamples to show the optimality of the embedding. In contrast to Littlewood’s theorem, random Bergman functions exhibit no improvement of regularity for any $p0$, but the loss of regularity for $p2$ is not as drastic as the Hardy case; there is indeed a nontrivial boundary curve given by $frac{1}{q}-frac{2}{p}+frac{1}{2}=0$. Several other results about random Bergman functions are established along the way. The technical difficulties, especially when $p1$, are different from the Hardy space and we devise a different route of proof. The Dirichlet space follows as a corollary. An improvement of the original Littlewood theorem is obtained.
Abstract. Let $f(z)=sum _{n=0}^{infty }a_n z^n$ be a formal power series with complex coefficients. Let $({mathcal{R}} f)(z)= sum _{n=0}^{infty }pm a_n z^n$ be the randomization of $f$ by choosing independently a random sign for each coefficient. Let $H^p({mathbb{D}})$ and $L^p_a({mathbb{D}})$ $(p0)$ denote the Hardy space and the Bergman space, respectively, over the unit disk in the complex plane. In 1930, Littlewood proved that if $f in H^2({mathbb{D}})$, then ${mathcal{R}} f in H^p({mathbb{D}})$ for any $p in (0, infty )$ almost surely, and if $f notin H^2({mathbb{D}})$, then ${mathcal{R}} f notin H^p({mathbb{D}})$ for any $p in (0, infty )$ almost surely. In this paper, we obtain a characterization of the pairs $(p, q) in (0, infty )^2$ such that ${mathcal{R}} f$ is almost surely in $L^q_a({mathbb{D}})$ whenever $f in L^p_a({mathbb{D}})$, including counterexamples to show the optimality of the embedding. In contrast to Littlewood’s theorem, random Bergman functions exhibit no improvement of regularity for any $p0$, but the loss of regularity for $p2$ is not as drastic as the Hardy case; there is indeed a nontrivial boundary curve given by $frac{1}{q}-frac{2}{p}+frac{1}{2}=0$. Several other results about random Bergman functions are established along the way. The technical difficulties, especially when $p1$, are different from the Hardy space and we devise a different route of proof. The Dirichlet space follows as a corollary. An improvement of the original Littlewood theorem is obtained.
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影响因子：
1.3
作者：
Fuchang Gao
通讯作者：
Fuchang Gao
DOI：
10.1007/bf02384861
发表时间：
1992-12
期刊：
Arkiv för Matematik
影响因子：
--
作者：
G. Bomash
通讯作者：
G. Bomash
DOI：
10.1515/9783110329841.559
发表时间：
2014-01
期刊：
USCO and Quasicontinuous Mappings
影响因子：
--
作者：
Dmitrii S. Silvestrov
通讯作者：
Dmitrii S. Silvestrov
DOI：
10.1090/surv/138
发表时间：
1990
期刊：
--
影响因子：
--
作者：
Kehe Zhu
通讯作者：
Kehe Zhu
影响因子：
1.3
作者：
A. Edelman;E. Kostlan
通讯作者：
A. Edelman;E. Kostlan