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

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 $H$ denot