Nearly-optimal estimates for the stability problem in Hardy spaces

TL;DR

This paper derives nearly-optimal bounds for the stability problem in Hardy spaces, quantifying how small boundary conditions on non-Blaschke sets influence the maximum of functions inside the disk.

Contribution

It provides new upper and lower bounds for the stability constant in Hardy spaces for non-Blaschke sets, extending previous work with nearly optimal estimates.

Findings

Bounds are nearly optimal for sets in compact subsets of the disk.

Bounds apply to sets in finite unions of Stolz angles.

Results generalize previous stability estimates in Hardy spaces.

Abstract

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 $\epsilon >0$ define C_p(\varepsilon, R) = \sup \{\sup_{|z| \leq R}|g(z)|: g\in H^p, \|g\|_p\leq 1, |g(\zeta)| \leq \varepsilon \forall \zeta\in E\}. In this paper we find upper and lower bounds for $C_p(\epsilon, R)$ when $\epsilon$ 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.