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arXiv · 2610.32922

A direct proof of the equivalence of the A2 and the Helson--Szego conditions

Christos Grigoriadis·Michail Paparizos·Georgios Psaromiligkos
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Abstract

We give a direct proof of the equivalence between the Muckenhoupt $\mathcal{A}_2$ and the Helson--Szegő conditions on the unit circle $T$. A positive weight $w$ belongs to $\mathcal{A}_2(T)$ if and only if $\log w=u+Hv$, where $u,v$ are real bounded functions, $\|v\|_\infty<\fracπ{2}$, and $H$ denotes the periodic Hilbert transform. The classical proof combines the Helson--Szegő theorem with the Hunt--Muckenhoupt--Wheeden characterization of weighted boundedness of the Hilbert transform. Our proof derives the Helson--Szegő representation directly from the Muckenhoupt $\mathcal{A}_2$ condition, without using boundedness of $H$ on $L^2(w)$.

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Christos Grigoriadis, Michail Paparizos, Georgios Psaromiligkos. 2026-10-23. A direct proof of the equivalence of the A2 and the Helson--Szego conditions. https://arxiv.org/abs/2610.32922

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