Research Problems in Function Theory — Problem 8.16

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.

For a domain $\Omega\subset\mathbb C^n$ equal to the unit polydisk $\mathbb D^n$ or the unit ball $\mathbb B^n$, define the Hardy class $H^p(\Omega)$ by \[\sup_{0<r<1}\int_{\partial\Omega}|f(r\zeta)|^p\,d\sigma(\zeta)<\infty,\] where $d\sigma$ is normalized product angular measure on the distinguished boundary $\mathbb T^n$ of the polydisk, and normalized surface measure on the sphere for the ball. Products mean single products $gh$, not sums of products.

Statement recorded in the supplied source.

Does there exist $f\in H^1(\mathbb D^2)$ that cannot be expressed as $gh$ with $g,h\in H^2(\mathbb D^2)$? The source update reports Rosay’s affirmative answer for $n=2$. It also asks the analogous question on $\mathbb B^n$ for every $n>1$: is there $f\in H^1(\mathbb B^n)$ that is not a product of two $H^2(\mathbb B^n)$ functions? [S2]

Short English statement

Record the failure-of-factorization question on the bidisk and its higher-dimensional ball analogue; the source says the bidisk case was settled.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308016 (AMR-022-8016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 8.16, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.