An estimate for weighted Hilbert transform via square functions (Q2781379)

Let \(A_2\) be the class of all \(A_2\)-weights, which means that a positive \(L^1_{\text{loc}}\) function \(\omega\in A_2\) if and only if \(Q_2(\omega):= \sup_I\langle\omega\rangle_I \langle\omega^{-1}\rangle_I< \infty\), where the supremum is taken over all intervals \(I\subseteq \mathbb{R}\), and the notion \(\langle \omega\rangle_I\) stands for the average of the function \(\omega\) over \(I\). In this paper, the authors prove that the Hilbert transform \(Hf(x)= \text{p.v. }\int_{\mathbb{R}} f(x- y) y^{-1} dy\) is a bounded operator on \(L^2(\omega)\) with the operator norm \(\|H\|\leq cQ_2(\omega)^{3/2}\) (the previous known result is \(\|H\|\leq cQ_2(\omega)^2\)). To prove the theorem, the authors reduce the problem to upper and lower bounds of certain square functions and use the averaging technique from [\textit{S. Petermichl}, C. R. Acad. Sci., Paris, Sér. I, Math. 330, No. 6, 455-460 (2000; Zbl 0991.42003)].