Let and be locally finite positive Borel measures on which do not share a common point mass. Assume that the pair of weights satisfy a Poisson condition, and satisfy the testing conditions below, for the Hilbert transform ,
with constants independent of the choice of interval . Then maps to , verifying a conjecture of Nazarov, Treil, and Volberg. The proof uses basic tools of nonhomogeneous analysis with two components particular to the Hilbert transform. The first component is a global-to-local reduction which is a consequence of prior work by Lacey, Sawyer, Shen, and Uriarte-Tuero. The second component, an analysis of the local part, is the particular contribution of this article.
Michael T. Lacey. "Two-weight inequality for the Hilbert transform: A real variable characterization, II." Duke Math. J. 163 (15) 2821 - 2840, 1 December 2014. https://doi.org/10.1215/00127094-2826799