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An estimate for weighted Hilbert transform via square functions. (English) Zbl 1005.42002
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 [S. Petermichl, C. R. Acad. Sci., Paris, Sér. I, Math. 330, No. 6, 455-460 (2000; Zbl 0991.42003)].

MSC:
 42A50 Conjugate functions, conjugate series, singular integrals 42A61 Probabilistic methods for one variable harmonic analysis
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References:
 [1] Stephen M. Buckley, Summation conditions on weights, Michigan Math. J. 40 (1993), no. 1, 153 – 170. · Zbl 0794.42011 [2] S. HUKOVIC, Thesis, Brown University, 1998. [3] S. Hukovic, S. Treil, and A. Volberg, The Bellman functions and sharp weighted inequalities for square functions, Complex analysis, operators, and related topics, Oper. Theory Adv. Appl., vol. 113, Birkhäuser, Basel, 2000, pp. 97 – 113. · Zbl 0972.42011 [4] F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909 – 928. · Zbl 0951.42007 [5] S. PETERMICHL Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, Comptes Rendus Acad. Sci. Paris, t.330, no.6, pp. 455-460, 2000. · Zbl 0991.42003 [6] S. PETERMICHL A sharp estimate of the weighted Hilbert transform via classical $$A_p$$ characteristic, Preprint, Insitute of Advanced Studies, 2001. [7] Janine Wittwer, A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7 (2000), no. 1, 1 – 12. · Zbl 0951.42008
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