Research Problems in Function Theory — Problem 5.27

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $D\subset\mathbb C$ be an arbitrary connected open set supporting a nonconstant bounded holomorphic function. Write $H^\infty(D)$ for its bounded holomorphic functions.

Statement recorded in the supplied source.

For every integer $n\ge1$, every $\delta>0$, and every $f_1,\ldots,f_n\in H^\infty(D)$ satisfying \[\sum_{j=1}^n|f_j(z)|\ge\delta\qquad(z\in D),\] must there exist $g_1,\ldots,g_n\in H^\infty(D)$ with $\sum_{j=1}^n f_jg_j\equiv1$ on $D$? The question is for all such plane domains, including infinitely connected ones. [S2]

Short English statement

Does the corona equation have bounded holomorphic solutions on every plane domain that supports nonconstant bounded holomorphic functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305027 (AMR-022-5027). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.27, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.