Two-weight inequality for the Hilbert transform: A real variable characterization, I

Let σ and w be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a Poisson A2 condition, and satisfy the testing conditions below, for the Hilbert transform H, ∫IH(σ1I)2dw≲σ(I),∫IH(w1I)2dσ≲w(I), with constants independent of the choice of interval I. Then H(σ⋅) maps L2(σ) to L2(w), verifying a conjecture of Nazarov, Treil, and Volberg. The proof has two components, a …