Research Problems in Function Theory — Problem 8.21

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 $X$ be the circle with arc-length measure or the real line with Lebesgue measure, and let $w\ge0$ be integrable on $X$. Write \[[w]_{A_2}=\sup_I\left(\frac1{|I|}\int_Iw\right)\left(\frac1{|I|}\int_Iw^{-1}\right),\] where $I$ ranges over nondegenerate arcs or finite intervals and nonintegrable reciprocals give value $+\infty$. The Helson--Szegő condition is $\log w=u+Hv$ almost everywhere, where $u,v$ are real bounded functions with $\|v\|_\infty<\pi/2$, and $H$ is the conjugate-function (Hilbert-transform) operator: on the circle, the principal-value kernel is $(2\pi)^{-1}\cot((\theta-t)/2)$, while on the line it is $\pi^{-1}(x-t)^{-1}$, with the usual subtraction at infinity and harmless additive constants absorbed into $u$.

Statement recorded in the supplied source.

Prove that $[w]_{A_2}<\infty$ is equivalent to the Helson--Szegő condition without using the boundedness criterion \[\int_X|Hf|^2w\le C\int_X|f|^2w\] as an intermediate characterization. [S2]

Short English statement

Give a direct proof of the equivalence between the A2 weight condition and Helson–Szegő factorization, avoiding the weighted Hilbert-transform criterion.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308021 (AMR-022-8021). 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 8.21, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.