Hardy空间中稳定性问题的近似最优估计
Nearly-optimal estimates for the stability problem in Hardy spaces
摘要：
我们继续开展\cite{TLNT}的工作。设$E$是复平面$\mathbb{C}$的单位圆$\mathbb{D}$的非Blaschke子集。固定$1\leq p\leq \infty$，设$H^p(\mathbb{D})$是圆盘上边值函数在$L^p(\partial \mathbb{D})$中的全纯函数的Hardy空间.已修复$0<R<1$。对于$ε>0$，定义C_p（\varepsilon，R）= \sup\{\sup_{|z| \leq R}| g（z）|：g \in H ^p，\| g\|_p \leq 1，|g（单位：千克）|\leq \varepsilon \forall和\in E。在本文中，我们找到了上界和下界为$C_p(ε, R)$时，$ε$是小的任何非Blaschke集$E$。的界限是接近最佳的许多这样的集合$E$，包括包含在一个紧凑的子集$\mathbb{D}$和包含在有限的工会Stolz角的集合。
英文摘要：
We continue the work of \cite{TLNT}. Let $E$ be a non-Blaschke subset of the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. Fixed $1\leq p\leq \infty$, let $H^p(\mathbb{D})$ be the Hardy space of holomorphic functions in the disk whose boundary value function is in $L^p(\partial \mathbb{D})$. Fixed $0<R<1$. For $ε>0$ define C_p(\varepsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(ζ)| \leq \varepsilon \forall ζ\in E\}. In this paper we find upper and lower bounds for $C_p(ε, R)$ when $ε$ is small for any non-Blaschke set $E$. The bounds are nearly-optimal for many such sets $E$, including sets contained in a compact subset of $\mathbb{D}$ and sets contained in a finite union of Stolz angles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
