An estimate for weighted Hilbert transform via square functions

Petermichl, S.; Pott, S.

doi:10.1090/S0002-9947-01-02938-5

An estimate for weighted Hilbert transform via square functions
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by S. Petermichl and S. Pott

Trans. Amer. Math. Soc. 354 (2002), 1699-1703

DOI: https://doi.org/10.1090/S0002-9947-01-02938-5

Published electronically: October 26, 2001

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Abstract:

We show that the norm of the Hilbert transform as an operator on the weighted space $L^2(w)$ is bounded by a constant multiple of the $3/2$ power of the $A_2$ constant of $w$, in other words by $c \sup _I (\langle \omega \rangle _I \langle \omega ^{-1} \rangle _I)^{3/2}$. We also give a short proof for sharp upper and lower bounds for the dyadic square function.

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