Nearly-optimal estimates for the stability problem in Hardy spaces

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.