Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón–Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.