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A direct proof of the equivalence of the A2 and the Helson--Szego conditions
Abstract
We give a direct proof of the equivalence between the Muckenhoupt 𝒜2 and the Helson--Szegő conditions on the unit circle T. A positive weight w belongs to 𝒜2(T) if and only if log w=u+Hv, where u,v are real bounded functions, |v|∞<π/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 𝒜2 condition, without using boundedness of H on L2(w).
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BibTeX
@misc{grigoriadis2026direct,
title = {{A direct proof of the equivalence of the A2 and the Helson--Szego conditions}},
author = {Christos Grigoriadis and Michail Paparizos and Georgios Psaromiligkos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32922},
eprint = {2610.32922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32922},
}
RIS
TY - UNPB
TI - A direct proof of the equivalence of the A2 and the Helson--Szego conditions
AU - Christos Grigoriadis
AU - Michail Paparizos
AU - Georgios Psaromiligkos
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32922
UR - https://arxiv.org/abs/2610.32922
AB - We give a direct proof of the equivalence between the Muckenhoupt 𝒜_2 and the Helson--Szegő conditions on the unit circle T. A positive weight w belongs to 𝒜_2(T) if and only if log w=u+Hv, where u,v are real bounded functions, |v|_∞<π/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 𝒜_2 condition, without using boundedness of H on L^2(w).
ER -