Research Problems in Function Theory — Problem 7.81

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 $H^\infty=\mathcal O(\mathbb D)\cap L^\infty(\mathbb D)$ with norm $\|f\|_\infty=\sup_{z\in\mathbb D}|f(z)|$, and let $(H^\infty)^*$ be its bounded complex-linear dual. An inner function is an element of $H^\infty$ with radial boundary values of modulus $1$ almost everywhere. Let $I$ be all such functions, including unimodular constants; the dual norm is $\|\Lambda\|=\sup_{\|f\|_\infty\le1}|\Lambda(f)|$.

Statement recorded in the supplied source.

If $\Lambda_n\in(H^\infty)^*$ and for every $\phi\in I$ there is a finite $C(\phi)$ with $|\Lambda_n(\phi)|\le C(\phi)$ for all $n$, must $\sup_n\|\Lambda_n\|<\infty$? [S2]

Short English statement

Does pointwise boundedness on all inner functions force uniform boundedness of a sequence of bounded linear functionals on the bounded holomorphic functions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307081 (AMR-022-7081). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.81, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.