An estimate for weighted Hilbert transform via square functions

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.