Research Problems in Function Theory — Problem 7.35

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.

On the unit circle use angular Lebesgue measure. A real integrable $u$ is in $\operatorname{BMO}$ if \[\sup_I\frac1{|I|}\int_I|u-u_I|\,d\theta<\infty,\qquad u_I=|I|^{-1}\int_Iu\,d\theta, \] where $I$ ranges over arcs. For a real bounded function $b$, its conjugate $\widetilde b$ is the circular Hilbert transform with zero mean: its Fourier coefficient of index $k\ne0$ is $-i\operatorname{sgn}(k)\widehat b(k)$ and its coefficient of index zero is zero. Equalities of boundary functions are taken almost everywhere and $\|\cdot\|_\infty$ is the essential supremum.

Statement recorded in the supplied source.

Given real $u\in\operatorname{BMO}$, determine the optimal value \[\inf\{\|b_2\|_\infty:u=b_1+\widetilde b_2,\quad b_1,b_2\text{ real and bounded}\}.\] [S2] Fefferman's decomposition assures admissible pairs exist. The original relates the optimisation to Baernstein's zero-free univalent-function factorisation question in Problem5.58. [S2]

Short English statement

Given a real function of bounded mean oscillation, determine the smallest possible size of its bounded conjugate component in a bounded-plus-conjugate decomposition.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307035 (AMR-022-7035). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.35 and its complete update; Problem5.58 and its update. https://arxiv.org/abs/1809.07200.