Research Problems in Function Theory — Problem 5.63

Definitions and mathematical statement

Let $\mathbb D$ be the unit disc and $\mathbb T$ its boundary with normalized arclength measure. The space BMOA consists of holomorphic functions $f\in H^1$ whose almost-everywhere boundary function $f^*$ has bounded mean oscillation: \[ \sup_{I\subset\mathbb T}\frac1{|I|}\int_I|f^*-(f^*)_I|\,|d\zeta|<\infty, \quad (f^*)_I=\frac1{|I|}\int_I f^*\,|d\zeta|, \] where $I$ ranges over arcs. For $f\in\mathrm{BMOA}$ define \[ \nu(f)=\inf_{q\in\mathcal O(\mathbb D)} \sup_{z\in\mathbb D}|f(z)+\overline{q(z)}|. \] Here $\mathcal O(\mathbb D)$ denotes all holomorphic functions. Constants are immaterial because they can be cancelled by $q$, so this is a seminorm on functions and a norm on the corresponding quotient by constants. Must there exist a holomorphic $q$ attaining this infimum for which the bounded harmonic function $H=f+\overline q$ has boundary values satisfying \[ |H^*(\zeta)|=\nu(f)\quad\text{for almost every }\zeta\in\mathbb T? \] The request combines optimality with constant boundary modulus; an arbitrary bounded correction or an approximating minimizing sequence does not suffice. The source records an affirmative answer for rational $f$ and asks for general BMOA functions.

Short English statement

Does every BMOA function admit an optimal antiholomorphic correction whose boundary modulus is constant almost everywhere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305063 (AMR-022-5063), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.63. Original source indexed by AMR-022-5063. https://arxiv.org/abs/1809.07200.