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An estimate for weighted Hilbert transform via square functions

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 2 power of the A 2 constant of w, in other words by c sup I ( r I ω -1 I ) 3/2 . We also give a short proof for sharp upper and lower bounds for the dyadic square function.