Research Problems in Function Theory — Problem 7.41

Definitions and mathematical statement

Let $S=\{z_\nu:\nu\geq1\}\subset\mathbb D$ consist of distinct points satisfying $\sum_\nu(1-|z_\nu|)<\infty$. Let $B$ be their Blaschke product, with unimodular constant arbitrary; its zeros are exactly $S$, each simple. For an open set $U$, $H^\infty(U)$ is the space of bounded holomorphic functions on $U$, with one finite bound on all components, and \[ H^\infty(U)|_S=\{(h(z_\nu))_{\nu\geq1}:h\in H^\infty(U)\} \] when $S\subset U$.

Question (A. Stray). For every such sequence and every $t>0$, is \[ H^\infty(\mathbb D)|_S=H^\infty(W_t)|_S, \qquad W_t=\{z\in\mathbb C:|B(z)|<t\}, \] as printed? The definition is incomplete because the infinite Blaschke product is initially defined only in $\mathbb D$, with no global continuation specified.

Under the natural but additional in-disk interpretation \[ W_t=\{z\in\mathbb D:|B(z)|<t\}, \] the nontrivial direction says that every bounded holomorphic $h$ on $W_t$ has a bounded holomorphic interpolant $f$ on $\mathbb D$ with $f(z_\nu)=h(z_\nu)$ for every $\nu$. No uniform norm bound is requested by the displayed set equality. The reverse inclusion is restriction. However, for this interpretation $W_t=\mathbb D$ when $t\geq1$, whereas the source also says equality is already known for every Blaschke sequence when $0<t<1$. Those statements would cover all $t>0$, conflicting with the record's open status. No alternative global domain or corrected parameter interval is guessed.

The source separately reports the equality for interpolating sequences, defined by $H^\infty(\mathbb D)|_S=\ell^\infty$, via Earl's proof of Carleson's interpolation theorem, and attributes the stated $0<t<1$ general case to Carleson's original corona proof. These supplied claims are retained without resolving the domain/status inconsistency.

Short English statement

Compare bounded-holomorphic interpolation data on the disk with data on Blaschke sublevel sets. The global sublevel-set definition is missing, and the obvious in-disk repair makes the supplied known cases exhaust the question.

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 7.41, attributed to A. Stray, https://arxiv.org/abs/1809.07200; source-TeX extraction, accessed 2026-07-29. The 2018 update was retained for status triage.
• Earl's proof of Carleson's interpolation theorem; Carleson's original proof of the $H^\infty$ corona theorem, as named in the supplied statement. No publication identifiers, dates, or theorem locations are supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $2307041$, source code catalogue entry, Research Problems in Function Theory --- Problem 7.41. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.